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Studies on Integrability for Nonlinear Dynamical Systemsand its Applications

Koichi Kondo

Division of Mathematical Science

Department of Informatics and Mathematical Science

Graduate School of Engineering Science

Osaka University

Toyonaka, Osaka 560-8531, Japan

2001

Contents

List of Figures iii

List of Tables v

Chapter 1. Introduction 1

1. History of soliton theory 1

2. Integrability conditions 2

3. Integrable systems and numerical algorithms 5

4. Outline of the thesis 7

Chapter 2. Solution and Integrability of a Generalized Derivative Nonlinear Shrodinger

Equation 8

1. Introduction 8

2. Traveling wave solution 9

3. Painleve test 12

4. Numerical experiments 16

5. Concluding remarks 22

Chapter 3. An Extension of the Steffensen Iteration and Its Computational Complexity 25

1. Introduction 25

2. The Newton method and the Steffensen method 26

3. The Steffensen method and the Aitken transform 28

4. The Shanks transform and theε-algorithm 28

5. An extension of the Steffensen iteration 29

6. Convergence rate of the extended Steffensen iteration 30

7. Numerical examples and computational complexity 35


Chapter 4. Determinantal Solutions for Solvable Chaotic Systems and Iteration Methods

Having Higher Order Convergence Rates 42

1. Introduction 42

i

2. Trigonometric solutions for solvable chaos systems 43

3. The Newton method and the Nourein method 45

4. Addition formula for tridiagonal determinant 47

5. Determinantal solution for the discrete Riccati equation 49

6. Determinantal solutions for hierarchy of the Newton iteration 50

7. Determinantal solutions for hierarchy of the Ulam-von Neumann system 54

8. Determinantal solutions for hierarchy of the Steffensen iteration 58


Chapter 5. Concluding Remarks 63

Bibliography 66

List of Authors Papers Cited in the Thesis and Related Works 71

ii

List of Figures

2.1 Shape of traveling wave solution forα = 1, ω = 1/2 andV = 1/2. Solid line:

β = 0, D = 0.5. Dotted line:β = 0.91355, D = 0.00000976. Dashed line:

β = 0.9135528· · ·= (−1+√

31)/5, D = 0. 12

2.2 Behavior of solitary waves forα = 1, β = 0. 18

2.3 Behavior of solitary waves forα = 1, β = 0.5625. 19

2.4 Behavior of solitary waves forα = 1, β = 0.8. 20

2.5 Changes of the peaks of pulse-1 and pulse-2 after 10 times interactions. 21

2.6 Velocities of solitary waves after 10 times interactions. Circle: pulse-1,

triangle: pulse-2. Initial velocities are0.1 and−2.0, respectively. 22

2.7 Quantity of position shift (phase shift) per one interaction. Circle: pulse-1,

triangle: pulse-2. 23

2.8 Quantity of ripple generated after 10 times interactions.β v.s. ratio of

integrated values of ripple to the conserved quantity. Filled circle: the cases

of Eq. (2.28), circle: other cases. 24

3.1 Graphical explanation of the Steffensen iteration 27

3.2 A comparison oflog10| f (xn)| of several iteration methods whenφ ′(α), 0,±1.

(Example 1) Solid line: simple iteration. Dashed line: Steffensen iteration.

Circles, squares and triangles denote the extended Steffensen iteration for

k = 2,3, and4, respectively. 37

3.3 A comparison oflog10| f (xn)| of several iteration methods whenφ ′(α) = 0,

φ ′′(α) , 0. (Example 2) Dashed line: Newton method. Pluses, circles,

squares and triangles denote the extended Steffensen iteration fork = 1,2,3,

and4, respectively. 37

3.4 The parameters(l ,e) for which the Newton iterations do not converge.

(Example 3) 39

iii

3.5 The parameters(l ,e) for which the Steffensen iterations do not converge.

(Example 3) 39

3.6 The parameters(l ,e) for which the extended Steffensen iterations fork = 2 do

not converge. (Example 3) 40

4.1 Behavior of the Newton method (4.6) for the caseI = a0b0 > 0. 44

4.2 Behavior of the Newton method (4.6) for the caseI = a0b0 < 0. 44

iv

List of Tables

2.1 Fluctuation of the conserved quantity∆σ/σ . 17

3.1 Number of iterations and convergence rate. (Example 1) 36

3.2 Number of iterations and convergence rate. (Example 2) 38

3.3 Number of iterations and total numbers of mappings. (Example 3) 38

v

CHAPTER 1

Introduction

In this thesis, we study integrability for nonlinear dynamical systems including differential

equations and discrete equations based on the soliton theory. Furthermore, we study applica-

tions of the soliton theory to numerical algorithms.

1. History of soliton theory

The notion ofsolitonmeans the solitary wave that travels stably and preserves its shape after

interactions. The first literature about the soliton equations was presented in 1895 by Korteweg

and de Vries. They presented the differential equation

∂u∂ t

+u∂u∂x

+∂ 3u∂x3 = 0 (1.1)

which describes the propagation of a shallow water wave. The dispersion term∂ 3u/∂x3 causes

the wave to be scattered to many waves that have different phase velocities. The nonlinear

termu∂u/∂x varies the velocity of the wave according to the amplitude of the wave, then the

wave stands erect and soon collapses. From those reasons, it was believed that there did not exist

stable solitary wave for nonlinear evolution equations, until Korteweg and de Vries succeeded to

derive the equation that had the exact solution of solitary wave. From the balance of dispersion

and nonlinearity, the solution was obtained. The equation they presented is nowadays calledthe

KdV equation.

Although the KdV equation was discovered at early year, the next development of it had

not appeared until the research [89] by Zabusky and Kruskal in 1965. Using computers, they

simulated the KdV equation numerically. They set the initial condition as the superposition

of two pulses, both of which were exact solutions of solitary wave of the KdV equation. They

computed a time evolution of the waves with periodic boundary condition. Two pulses moved to

same direction by different velocities, because they had different amplitudes. The higher pulse

traveled faster than the lower one. Zabusky and Kruskal observed the behaviors of interactions

of pulses. From the results of the experiment, they discovered that each pulse preserved its shape

and its velocity after the interactions. Moreover they discovered that positions of pulses were

shifted at the interactions. That phenomenon is called a phase shift. Solitary waves behaved

1

like particles. Then they named such solitary wave as the ‘soliton’ (a suffix ‘-on’ stands for

a particle). Their numerical experiments found a new phenomenon for nonlinear evolution

equations. This discovery was also important as an example of contributions of computers to

developments of mathematics.

Next epoch-making discovery wasthe inverse scattering transform(IST) [23], which was

presented by Gardner, Greene, Kruskal, and Miura in 1967. By the IST, we transform a given

evolution equation to a certain linear integral equation. Then we can solve initial value problem

in principle. Another method for solving soliton equation was developed by Hirota in 1970s

(cf. [29]–[38], [43], [45]). It is calledHirota’s direct method.By the direct method, we can

solve soliton equation directly not via the IST. The direct method firstly transform a given

equation to so-calledHirota’s bilinear form. Then we exactly obtain exactN-soliton solution by

calculating a perturbation of the bilinear form. That solution is also expressed as a determinant.

Such determinantal solution is calledtheτ-function solution.And the bilinear form is reduced

to a certain identity of determinants.

The invention of the direct method also brought to us the techniques to discretize soliton

equations (cf. [39]–[42], [44]). Preserving the structure of theτ-function, we do discretize the

evolution equation, the independent variable transformation, the bilinear form, and the solu-

tion, simultaneously. Such discretization is sometimes calledan integrable discretization.For

example, the discrete KdV equation [39] is given by

ut+1n −ut−1

n+1 =1

utn+1

− 1ut

n. (1.2)

Many discrete soliton equations are now presented.

In early 1980s, Sato discovered that theτ-function of the Kadomtsev-Petviashvili (KP)

equation is closely related to algebraic identities such as determinant identities. Moreover, he

found that the totality of solutions for the KP equation and its higher order equations constitute

an infinite dimensional Grassmann manifold.

2. Integrability conditions

The notion of integrability is rigidly defined for Hamilton systems. If a Hamilton system

of N degree of freedom hasN independent and mutually involutive integrals, then the system

of ordinary differential equations (ODEs) is integrable in the sense in which the system can

be linearized in terms of successive canonical transformations. This is the main result in the

Liouville-Arnold theory. For partial differential equations (PDEs), there is no rigid definition

determined yet. However there are candidates for integrability conditions of those systems.

2

From studies on soliton equations, the following properties are now accepted as definitions of

integrability for PDEs.

(1) Solvability by IST.

(2) Existence ofN-soliton solution.

(3) Existence of infinite number of conserved quantities or symmetries.

(4) Existence of Lax pair [53].

(5) Existence of bilinear form.

Generally it is not easy to obtain explicit solutions and conserved quantities for a given

nonlinear equation. So we want to detect whether an equation is integrable or not beforehand.

Thus the following integrability criteria have been proposed:

(a) The Painleve test for ODE.

(b) The Weiss-Tabor-Carnevale (WTC) method for PDE.

(c) The singularity confinement test for discrete equation.

(d) The algebraic entropy test for discrete equation.

Those criteria are also used for deciding the values of parameters of an equation that has a

possibility of integrability. We shall briefly introduce them.

We first consider ODE. The singularities of a linear ODE all depend on coefficients of the

equation. However the singularities of a nonlinear equation often depend on initial values. We

here consider a simple example

dydx

+y2 = 0. (1.3)

The general solution of this equation is given by

y(x) =1

x−C. (1.4)

The singularity ofy(x) occurs atx=C. Since the constantC is determined byC =−1/y(0), the

singular point is moved according to the initial value. Such singular point is calleda movable

singular point. If any movable singular point of an equation is not critical point, namely all

movable singular points are poles, then it is called that the equation hasthe Painleve property.

The Painleve property is used for a criterion of integrability of ODE. We shall briefly review

the history of applications of the Painleve property.

In 1889, Kowalevskya presented a new integrable case of the rigid body about fixed point.

The equation of motion of the rigid body is sixth order ODE with six parameters. People at that

time knew that only two cases of the equations are integrable when the parameters are special-

ized as some values. Those equations are called Euler’s top and Lagrange’s top respectively. In

3

order to solve the equation, Kowalevskya restricted the solution to no movable singular point

except for movable poles. Under that condition, she specified the parameters and succeeded to

integrate the equation. The equation she presented is now called Kowalevskya’s top.

In 1900s, Painleve and co-workers presented so-calledthe Painleve equations.They inves-

tigated nonautonomous second order ODEs, and enumerated all equations that had no movable

critical point. They classified the equations and showed that the equations are essentially re-

duced to six types of new equations and known ones. Solutions of those six equations are called

the Painleve transcendents.

We here show how to check the Painleve property of a given ODE. Let a movable singularity

of y(x) occur atx = C. Then we expandy(x) around the pointx = C by the Laurent series

y(x) = (x−C)a∞

∑j=0

y j (x−C) j . (1.5)

We first check whether the singularity is a pole. It needs that the leading ordera is a finite

negative integer. If the leading order was a rational integer or an infinite integer, then the

singularity became a branch point or an essential singularity. Next we check that the Laurent

coefficientsy j have enough ambiguity. It needs that the number of arbitrary constants ofy j and

the initial constantC is the same as the time of differentiations of the equation. Ifa andy j

satisfy those conditions and the expansion has no inconsistency, then it is said that the equation

passes the Painleve test.

We next consider PDE case. A conjecture about integrability for PDE was proposed by

Ablowitz, Ramani, and Segur [1, 2, 3]. They stated that:

Every nonlinear ODE obtained by an exact reduction of a nonlinear PDE that is

solvable by IST has the Painleve property.

Many soliton solutions are known to have this property. The KdV equation is actually reduced

to an equation of elliptic function by a reduction of traveling wave solution. The modified KdV

equation is reduced to the Painleve equation of type II by a reduction using similarity solution.

However, it is impossible to check the Painleve property of all ODEs obtained by all re-

duction of a given PDE. Thus Weiss, Tabor, and Carnevale proposed a method to check the

Painleve property of PDE directly not via reductions. This method is calledthe WTC method

[84]. We briefly show the procedure of the WTC method. Let singularities of solutionu(x, t)for a nonlinear PDE occur on a manifoldφ(x, t) = 0. We assume that the functionφ(x, t) is an

arbitrary function, and that the solution is expressed as a formal Laurent series

u(x, t) = φ(x, t)a∞

∑j=0

u j(x, t)φ(x, t) j . (1.6)

4

We check that the leading ordera is a finite negative integer, and that the number of arbitrary

functions ofu j andφ is the same as the order of the differential equation. Ifa, u j andφ satisfy

those conditions and the expansion has no inconsistency, then it is said that the PDE has the

Painleve property. If it is necessary to restrictu j andφ to some conditions, then it is said that

the equation has the conditional Painleve property. An evolution equation that has a conditional

Painleve property is considered as a near-integrable system. In this thesis, we consider stability

of such an equation.

Next we consider discrete equation. A criterion for discrete systems was first proposed by

Grammaticos, Ramani, and Papageorgiou [25]. Their criterion is based on the property ofthe

singularity confinement(SC). The SC property means that:

The singularities of a discrete system are movable, i.e., they depend on initial

conditions. And the memory of the initial conditions survives past the singularity

by a few steps.

The property of the SC is accepted as a discrete version of the Painleve property. The discrete

Painleve equations and many discrete soliton equation pass the SC test.

The SC test has been a useful criterion. However, Hietarinta and Viallet presented an equa-

tion that passes the SC test but has numerically chaotic property [28]. Then they proposed a

more sensitive criterion. Their criterion is based onthe algebraic entropythat is defined by the

logarithmic average of a growth of degrees of iterations. The algebraic entropy test and the SC

test are similar to each.

The SC type criteria are effective in reversible discrete systems such as soliton equations.

However they are ineffective in irreversible discrete systems. For example, the arithmetic-

harmonic mean algorithm [62],

an+1 =an +bn

2, bn+1 =

2anbn

an +bn, (1.7)

has the explicit solution, however does not pass the SC test. We consider in the thesis integra-

bility of such equations.

3. Integrable systems and numerical algorithms

The soliton theory has been developed in mathematics, physics and engineering. The op-

tical soliton communication [26] is a famous example of application of the soliton theory to

communication engineering. There are also applications to mathematical engineering. A close

relationship between soliton equations and numerical algorithms has been pointed out. We

enumerate those numerical algorithms and related integrable systems as follows.

5

• Matrix eigenvalue algorithms

– 1-step of the QR algorithm is equivalent to time1 evolution of the ordinary Toda

equation [75] (see [73]).

– The LR algorithm is equivalent to the discrete Toda equation [40] (see [46]).

– The power method with the optimal shift is derived from an integrable discretiza-

tion of the Rayleigh quotient gradient system (see [60]).

• Convergence acceleration algorithms

– The recurrence relation of theε-algorithm [85] (cf. the Shanks transform [70]) is

equivalent to the discrete potential KdV equation (see [68]).

– Theρ-algorithm [86] is equivalent to the discrete cylindrical KdV equation (see

[68]).

– Theη-algorithm is equivalent to the discrete KdV equation (see [56]).

– Then-th term of theE-algorithm is equivalent to the solution of the discrete hun-

gry Lotka-Volterra equation (see [76]).

• Continued fraction algorithms (Pade approximations)

– The recurrence relation of the qd algorithm for calculating continued fraction is

equivalent to the discrete Toda equation.

– The ordinary Toda equation gives a method for calculating Laplace transforms via

the continued fraction (see [61]).

– A new Pade approximation algorithm is formulated by using the discrete Schur

flow (see [55]).

• Decoding algorithms

– A BCH-Goppa decoding algorithm is designed by the Toda equation over finite

fields (see [59]).

• Iteration methods having higher order convergence rate

– The recurrence relation of the arithmetic-geometric mean algorithm has the solu-

tion of theta function (see [18]).

– The recurrence relation of the arithmetic-harmonic mean algorithm has the solu-

tion of hyperbolic function (see [62]).

From these results, one may conjecture that a good numerical algorithm is regarded as

an integrable dynamical system. Indeed, eigenvalue algorithms and acceleration algorithms,

which are essentially linear convergent algorithms, pass the SC test of integrability criterion

(cf. [68]). Moreover, they are proved to be equivalent to discrete soliton equations via Hirota’s

bilinear forms. However, some algorithms having higher order convergence rate do not pass

this integrability criterion, as we mentioned in the previous section. It needs more discussions

6

about integrability for such equations. We consider integrability of algorithms in the thesis.

Furthermore, we develop numerical algorithms using the techniques in the soliton theory.

4. Outline of the thesis

The thesis is organized as follows.

In Chapter 2, we consider a generalized derivative nonlinear Schrodinger (GDNLS) equa-

tion. The equation is derived by adding two dispersion terms to the nonlinear Schrodinger

(NLS) equation [51, 26], which describes a propagation of pulses in optical fibers. The GDNLS

equation has two parameters. We first construct a traveling wave solution for arbitrary values

of parameters. We next investigate integrability of the GDNLS equation by the WTC method

of the Painleve test. We show that the equation has the Painleve property and a conditional

Painleve property for some conditions of parameters. By numerical experiments, we examine

stability of the traveling wave solutions in interactions.

In Chapter 3, we consider an extension of the Steffensen method [72]. The Steffensen

method is an iteration method for finding a root of nonlinear equations. Its iteration function is

constructed without any derivative function, and it has the second order convergence rate. The

point to devise our extended method is that the iteration function is defined by using thek-th

Shanks transform which is a sequence convergence acceleration algorithm. The convergence

rate is shown to be of orderk+ 1. The use of theε-algorithm avoids the direct calculation of

Hankel determinants, which appear in the Shanks transform, and then diminishes the compu-

tational complexity. For a special case of the Kepler equation, it is shown that the numbers of

mappings are actually decreased by the use of the extended Steffensen iteration.

In Chapter 4, we give new determinantal solutions for irreversible discrete equations. The

equations considered are solvable chaotic systems and the discrete systems which are derived

from iteration methods having higher order convergence rates. We deal with the hierarchy of the

Newton type iterations (the Newton method and Nourein method [64]), that of the Steffensen

type iterations (the Steffensen method and the extended Steffensen method in Chapter 3), and

that of the Ulam-von Neumann system [77]. We obtain determinantal solutions for those sys-

tems including solvable chaotic systems in terms of addition formulas derived from some linear

systems.

In Chapter 5, we finally state some remarks and further problems.

7

CHAPTER 2

Solution and Integrability of a Generalized Derivative Nonlinear

Shrodinger Equation

1. Introduction

In this chapter, we consider the following equation,

iUt +12

Uxx+ |U |2U + iα |U |2Ux + iβ U2U∗x = 0, (2.1)

whereU = U(x, t) is a complex variable and∗ denotes a complex conjugate. Moreover,α and

β are real parameters. Eq. (2.1) is reduced to the well-known nonlinear Schrodinger (NLS)

equation

iUt +12

Uxx+ |U |2U = 0 (2.2)

for α = β = 0. Moreover, Eq. (2.1) yields two types of derivative nonlinear Schrodinger equa-

tions which are known to be integrable, namely the case ofα : β = 1 : 0 [58]

iUt +12

Uxx+ |U |2U + i |U |2Ux = 0, (2.3)

and the case ofα : β = 2 : 1 [83]

iUt +12

Uxx+ |U |2U +2i |U |2Ux + iU2U∗x = 0. (2.4)

Hereafter we call Eq. (2.1) a generalized derivative nonlinear Schrodinger (GDNLS) equation.

We note that the GDNLS equation (2.1) can be regarded as a special case of the higher order

nonlinear Schrodinger equation proposed by Kodama and Hasegawa [51]

iUt +12

Uxx+ |U |2U + iα |U |2Ux + iβ U2U∗x + iγ Uxxx = 0 (2.5)

which describes the pulses in optical fibers.

It is remarked that the term|U |2U can be eliminated by a gauge transformation [49]. Eqs. (2.3)

and (2.4) without this term are known as the Chen-Lee-Liu (CLL) equation [15] and the Kaup-

Newell (KN) equation [50], respectively. The CLL equation was discussed by using the bilinear

formalism by Nakamura and Chen [58]. Hirota [47] bilinearized the KN equation and showed

8

that the CLL equation and the KN equation have the same bilinear forms. A class of solutions

for the CLL, KN equations and their integrable generalization by Kundu [52]

iUt +12

Uxx+2iγ |U |2Ux +2i(γ−1)U2U∗x +(γ−1)(γ−2) |U |4U = 0, (2.6)

whereγ is a real parameter, has been constructed explicitly through the bilinear formalism by

Kakei et al. [49].

We first construct a traveling wave solution of the GDNLS equation (2.1) in Section 2. Moti-

vated by a concrete form of the solution, we investigate the integrability of the GDNLS equation

by using the Painleve test in Section 3. Finally we examine a behavior of the traveling wave

solution numerically in Section 4. In Section 5, we mention several remarks of this chapter.

2. Traveling wave solution

In this section, we construct a traveling wave solution for the GDNLS equation. Here we

remark that the values of parametersα , β in Eq. (2.1) are taken to be arbitrary by the scale

change except for the ratioβ/α , and henceβ/α can be regarded as a characteristic parameter

of the equation.

Eq. (2.1) is invariant under the following transformation

U(x, t) =1√kU(X,T)e−iV(X−V

2 T) , (2.7)

where

x = k(X−VT) , t = k2T , k = 1−Vα +Vβ (2.8)

andV is an arbitrary constant. Taking this invariance into account, we first construct a stationary

solution. We put

U(x, t) := r(x)exp(i θ(x))exp(i ω t) , (2.9)

wherer(x) andθ(x) are real functions inx, andω is a real constant. Substituting (2.9) into

Eq. (2.1), we get

rxx = 2ωr−2r3 + rθ 2x +2(α−β )r3θx (2.10)

from the real part and

θxx =−2rxθx

r−2(α +β )rr x (2.11)

9

from the imaginary part, respectively. The following ansatz is crucial for our construction of

solution

θx = κ r , (2.12)

whereκ is a constant. We obtain from Eq. (2.11)

(2κ +α +β )rr x = 0, (2.13)

from which we have

κ =−α +β2

. (2.14)

Then Eq. (2.10) becomes

rxx = 2ω r−2r3− 14(α +β )(3α−5β ) r5. (2.15)

Integrating Eq. (2.15), we obtain

r2 =8ω e±2

√2ω x

1+2e±2√

2ω x +(1+ 2

3ω(α +β )(3α−5β ))

e±4√

2ω x. (2.16)

Moreover, we get from Eqs. (2.12), (2.14) and (2.16)

θ =−√

3(α +β )3α−5β

tan−1

1+

(1+ 2

3ω(α +β )(3α−5β ))

e±2√

2ω x

√1+ 2

3ω(α +β )(3α−5β )

, (2.17)

where the following conditions should be satisfied that

ω ≥ 0, 1+23

ω(α +β )(3α−5β )≥ 0 (2.18)

for the reality ofr andθ . Substituting Eqs. (2.16) and (2.17) into Eq. (2.9), we have the station-

ary wave solution. Then, applying the transformation (2.7)–(2.8), we obtain the traveling wave

solution. The result is expressed as

U(x, t) =p+ p∗

e−φ +Qeφ∗

(e−φ +Qeφ∗

e−φ +Peφ∗

)N+12

. (2.19)

Here we define functionφ(x, t) as

φ = px+12

ip2t +φ (0) . (2.20)

And we define parametersp, Ω, P, Q, andN as

p = (1−Vα +Vβ )Ω+ iV , (2.21)

Ω =±√

2ω , (2.22)

10

P = (1−Vα +Vβ )

(1+Ω

√−1

3(α +β )(3α−5β )

), (2.23)

Q = (1−Vα +Vβ )

(1−Ω

√−1

3(α +β )(3α−5β )

), (2.24)

N =

√3(α +β )3α−5β

(2.25)

andφ (0) andV are arbitrary constants. The condition (2.18) is also necessary here. This solution

is characterized by the parametersω andV for fixed α andβ . The shape of the solution varies

by the value of

D = PQ= (1−Vα +Vβ )2

1+23

ω(α +β )(3α−5β )

. (2.26)

In fact, |U | is given by

|U |=√

(p+ p∗)2eφ+φ∗

1+2eφ+φ∗ +De2(φ+φ∗) . (2.27)

If D is sufficiently large, the solution has the soliton-like shape. ForD ∼ 0, it becomes trape-

zoidal shape and forD → 0, it has the kink-like shape as illustrated in Figure 2.1. Hence the

traveling wave solution (2.19) may behave as a solitary wave.

Here we remark that if we take the limitα,β → 0, this solution is reduced to the 1-soliton

solution of the NLS equation. Similarly, in the cases ofβ/α = 0 andβ/α = 1/2, it gives the

1-soliton solution of Eqs. (2.3) and (2.4), respectively.

It should be emphasized that ifN, which depends only on the ratio ofα andβ , is an odd

integer, the traveling wave solution (2.19) is rational in exponential functions which is the com-

mon feature of soliton solutions. Thus it might be expected that in such cases, solitary waves of

the GDNLS equation has good properties like that of integrable cases. The ratio ofα andβ in

such cases are given by

βα

=3m(m+1)

5m(m+1)+2, m= 0,1,2, . . . . (2.28)

The casesm= 0 andm= 1 correspond to Eqs. (2.3) and (2.4), respectively, and they are known

to be integrable as mentioned in the introduction.

Moreover, it should be noted that the 1-soliton solution for Eq. (2.6) obtained by Kakei et

al. [49] has a quite similar form to Eq. (2.19). Indeed, we can check that 1-soliton solution of

equation (2.6) forγ = 0,1 equivalent to (2.19) forN = 1,3, respectively, and not for other cases.

11

−20.0 −10.0 0.0 10.0 20.0x

0.0

0.5

1.0

1.5|U

|

FIGURE 2.1. Shape of traveling wave solution forα = 1, ω = 1/2 andV = 1/2.

Solid line: β = 0, D = 0.5. Dotted line:β = 0.91355, D = 0.00000976. Dashed

line: β = 0.9135528· · ·= (−1+√

31)/5, D = 0.

From the observation above, it may be natural to ask whether the cases ofm> 1 in Eq. (2.28)

are integrable or not. As for the integrability in a strict sense, the answer is no. In fact, Clarkson

and Cosgrove [17] investigated the Painleve property to the following equation,

i ut +uxx+ iα uu∗ux + iβ u2u∗x + γ u3u∗2 +δ u2u∗ = 0, (2.29)

whereα, β , γ , andδ are real parameters, and shown that it is integrable only the case when

it is equivalent to Eq. (2.6). However, we may expect some information from integrability test

which distinguish the cases of Eq. (2.28) from other cases. We consider the integrability of the

GDNLS equation (2.1) in the next section.

3. Painleve test

In this section, we investigate the integrability of the GDNLS equation by using so-called

the Painleve test proposed by Weiss et al. [84], and show that the GDNLS equation possesses

“conditional Painleve property” for the cases of Eq. (2.28).

12

Following to the procedure of the test, we regard thatu = U andv = U∗ are independent,

and consider the GDNLS equation as a coupled system

i ut +12

uxx+u2v+ iα uuxv+ iβ u2vx = 0, (2.30)

−i vt +12

vxx+v2u− iα vvxu− iβ v2ux = 0. (2.31)

We assume the formal Laurent expansion around the zero points of some analytic function

φ(x, t) for the solution of Eqs. (2.30) and (2.31)

u = φa∞

∑j=0

u j φ j , v = φb∞

∑j=0

v j φ j . (2.32)

In this method, if

(1) there is no movable critical points, namely, the leading ordersaandbare finite integers,

(2) the expansion (2.32) has sufficient number of arbitrary functionsu j andv j ,

(3) there is no incompatibilities in the expansion,

then it is regarded that the equation passes the Painleve test, or it is said that the equation pos-

sesses the Painleve property. In such case, it is usually believed that the equation is integrable.

We show the concrete analysis in the following.

3.1. Leading order analysis.To get the leading powera andb, we substituteu∼ u0φx

andv∼ v0φb into Eqs. (2.30) and (2.31). We obtain the relation

a+b =−1, (2.33)

to adjust the leading order, and find

a =12

(−1±

√3(α +β )3α−5β

), (2.34)

u0v0 =±i φx

√3

(α +β )(3α−5β ). (2.35)

Sincea andb should be integers, we get the condition

βα

=3m(m+1)

5m(m+1)+2, m= 0,1,2, . . . (2.36)

which is exactly the same as the condition (2.28).

13

3.2. Resonance analysis.The degreej is calledresonancewhenu j or v j becomes an ar-

bitrary function. The recurrence relation foru j andv j is given by(

A( j)11 A( j)

12

A( j)21 A( j)

22

)(u j

v j

)=

(Fj

G j

), j = 0,1,2,3, . . . . (2.37)

Here we define elementsA( j)11 , A( j)

12 , A( j)21 , andA( j)

22 as

A( j)11 =

12( j +a−1)( j +a)φ2

x + iα( j +2a)+2βbu0v0φx , (2.38)

A( j)12 = iαa+β ( j +b)u2

0φx , (2.39)

A( j)21 =−iαb+β ( j +a)v2

0φx , (2.40)

A( j)22 =

12( j +b−1)( j +b)φ2

x − iα( j +2b)+2βau0v0φx . (2.41)

And we defineFj andG j as some polynomials ofu j , v j andφ such that

Fj = Fj(u0, . . . ,u j−1,v0, . . . ,v j−1,φ) , (2.42)

G j = G j(u0, . . . ,u j−1,v0, . . . ,v j−1,φ) . (2.43)

Moreover we defineu j = v j = 0 for j < 0.

We shall obtain the resonances. Coefficientu j or v j can be an arbitrary function when the

condition

det

(A( j)

11 A( j)12

A( j)21 A( j)

22

)=

14

φ4x ( j +1) j( j−2)( j−3) = 0 (2.44)

is satisfied. Hence we find that the resonances are

j =−1,0,2,3. (2.45)

3.3. Compatibility condition. If the degreej is a resonance, the recurrence relation (2.37)

should satisfy the compatibility condition

A( j)11 : A( j)

21 = A( j)12 : A( j)

22 = Fj : G j (2.46)

or

Fj = 0, G j = 0. (2.47)

We shall check the compatibility for each resonance. Resonancej = −1 corresponds to the

arbitrariness ofφ . The compatibility condition is not necessary forj = −1. When j = 0, we

14

haveF0 = G0 = 0. When j = 2, we next obtain the relation

A(2)11

A(2)21

=A(2)

12

A(2)22

=F2

G2=± 2(2m+1)αu0

2

(5m2 +5m+2)iφx. (2.48)

Thus we have checked the compatibility for the resonancesj = 0,2. The resonancej = 0

corresponds to the arbitrariness ofu0 or v0, and j = 2 to that ofu2 or v2. For j = 3, if the

condition

v0(m+1)(m−1)φ4

x (2m+1)(2φtxφtφx−φttφ2

x −φ2t φxx) = 0, (2.49)

or

u0(m+2)mφ4

x (2m+1)(2φtxφtφx−φttφ2

x −φ2t φxx) = 0, (2.50)

is satisfied, then it is shown that the expansion is compatible. Therefore, form= 0 and1, the

compatibility conditions are automatically satisfied. However, form= 2,3,4, . . ., the function

φ(x, t) should satisfy

2φtxφtφx−φttφ2x −φ2

t φxx = 0 (2.51)

to pass the test.

From this result, we may conclude that the GDNLS equation (2.1) possesses the Painleve

property for the cases ofm= 0 and1 in Eq. (2.28) which are known to be integrable. Form> 1,

it does not pass the test in strict sense, but possesses “conditional Painleve property” [87, 88].

For other cases, it does not pass the test.

It may be interesting to remark here that the condition (2.51) yields the dispersionless KdV

equation

ft − f fx = 0 (2.52)

by the dependent variable transformation

f =φt

φx. (2.53)

We also note that exactly the same condition has appeared in the analysis of some system

which describes the interaction of long and short water waves [87, 88]. In [87, 88], Yoshinaga

conjectured that the equation which passes the Painleve test with the condition (2.51) has “finite-

time integrability”, since the solution of Eq. (2.52) loses analyticity in finite time as is well-

known, and thus the assumption of the Painleve test breaks.

15

4. Numerical experiments

4.1. Purpose.From the result of the Painleve test, the GDNLS equation is not integrable

in strict sense except for the casesm= 0 and1 in Eq. (2.28). However, from the structure of

the traveling wave solution, one may expect that the solitary waves behave like solitons even if

the equation itself is not integrable. Motivated by this, we numerically solve the initial value

problem for the GDNLS equation to check the following points:

(1) Stability of solitary waves in interactions.

(2) Existence of phase shift.

(3) Quantity of ripple which is generated by interactions.

(4) Any phenomenon which implies “finite-time integrability.”

If (1) and (2) are observed, then it can be said that the solitary waves behave like solitons. We

investigate (3) from the following reason: Suppose we observe the interaction of two different

solitary waves. If the equation has a 2-soliton solution, it must approximate the initial state well

at somet with some values of parameters. Then we may expect that the ripple which emerges

through the interaction is quite small. Conversely, if the ripple which is observed for some val-

ues ofα andβ is small compared to other cases, then we may expect the existence of 2-soliton

solution, or at least, it may be worth in further analysis. Moreover, it might be interesting to

check whether the behavior of solutions differs or not by the cases that the GDNLS equation has

the Painleve property, the conditional Painleve property and the other cases. From theoretical

point of view,β/α = 0.6 might be a critical point, since if the GDNLS equation possesses the

conditional Painleve property, thenβ/α should satisfy0≤ β/α < 0.6 from Eq. (2.36).

4.2. Method of numerical experiments.We adopt the spectral method for space, and the

Runge-Kutta method for time integration. Range in space is from−50 to 50 and the number

of mesh is29 = 512 points. Time interval is taken to be0.01. We take superposition of two

different traveling wave solutions as the initial value and calculate their time evolution. These

two solitary waves are put with sufficient distance att = 0. Then we fix the value ofα as1, and

examine the time evolution with different values ofβ . The values of characteristic parameters

of the traveling wave solutions are given byω = 0.55 andV = 0.1 for one wave,ω = 0.0075

andV =−2.0 for another wave, respectively. Hereafter we call the former solitary wave pulse-1

and the latter pulse-2.

4.3. Results.Calculations have been performed until the solitary waves interact 10 times.

We have checked the conserved quantityσ =∫|U |2dx during the calculation as a measure of

16

reliability. We see thatσ is kept with sufficient accuracy. In fact, fluctuation ofσ during the

calculation is at most∆σ/σ ∼ 10−8, as shown in Table 2.1.

TABLE 2.1. Fluctuation of the conserved quantity∆σ/σ .

t β = 0 β = 916 = 0.5625 β = 0.8

42 0 0 1.6755916945128×10−8

89 9.3196098838920×10−10 1.9700283767298×10−9 1.9287796510011×10−8

136 1.9166390124162×10−9 2.7629271109169×10−9 2.0984692295199×10−8

230 3.9552707787706×10−9 4.2005398496527×10−9 2.3408647116396×10−8

466 8.8452799018246×10−9 7.6076397313715×10−9 2.7791032609452×10−8

Figures 2.2 and 2.3 shows the behavior of solitary waves for the integrable caseβ = 0, and

the case ofβ = 0.5625= 9/16 (m = 1 in Eq. (2.28)), respectively. For smallβ , the solitary

waves are stable in interaction. Asβ becomes larger, the change of the shape of solitary waves

becomes large, which is illustrated in Figure 2.4.

Changes of heights of peaks and velocities for solitary waves after 10 times interactions for

differentβ are shown in Figures 2.5 and 2.6, respectively.

Phase shifts in interaction are also observed for anyβ as shown in Figure 2.7. We note that

the amounts of phase shift are measured by average of 10 times interactions.

Figure 2.8 shows the quantity of ripple after 10 times interactions of solitary waves. Here,

it is measured by the ratio of integrated values of ripple to the conserved quantityσ . We see

that the ripple is quite small for the integrable cases (β = 0,0.5), as was expected. But it looks

that it does not differ by the cases that the equation has the conditional Painleve property (filled

circles in Figure 2.8), and other cases (circles in Figure 2.8).

We have mentioned thatβ = 0.6 might be a critical point, but it looks that there is no drastic

change in behavior of solitary waves atβ = 0.6.

From these results, we may conclude that solitary waves are stable and behave like solitons

at least for smallβ . Difference of behavior between the cases that the GDNLS equation has

the conditional Painleve property and the other cases was not observed in our calculations. In

other words, we may conclude that soliton-like behavior of the solitary waves is the common

property of the GDNLS equations regardless of the parameterβ/α, as far as it is small.

As for the “finite-time integrability,” we could not observe any such phenomenon that im-

plies “finite-time integrability,” e.g., break down of solitary waves in our numerical calculations.

17

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.2

0.5

0.8

1.0|U

|

pulse−1

pulse−2

t = 0

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.2

0.4

0.6

0.8

1.0

|U|

t = 466

pulse−1

pulse−2

FIGURE 2.2. Behavior of solitary waves forα = 1, β = 0.

18

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.5

1.0

1.5|U

|t = 0

pulse−1

pulse−2

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.5

1.0

1.5

|U|

t = 466

pulse−1

pulse−2

FIGURE 2.3. Behavior of solitary waves forα = 1, β = 0.5625.

19

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.5

1.0

1.5|U

| pulse−1

pulse−2

t = 0

−50.0 −25.0 0.0 25.0 50.0x

0.0

0.5

1.0

1.5

|U|

t = 466

pulse−1

pulse−2

FIGURE 2.4. Behavior of solitary waves forα = 1, β = 0.8.

20

0.0 0.2 0.4 0.6 0.8β

−2.0

−1.0

0.0

1.0

2.0

[%]

pulse−1

0.0 0.2 0.4 0.6 0.8β

−2.0

−1.0

0.0

1.0

2.0

[%]

pulse−2

FIGURE 2.5. Changes of the peaks of pulse-1 and pulse-2 after 10 times interactions.

21

0.0 0.2 0.4 0.6 0.8β

−3.0

−2.0

−1.0

0.0

1.0ve

loci

ty

FIGURE 2.6. Velocities of solitary waves after 10 times interactions. Circle:

pulse-1, triangle: pulse-2. Initial velocities are0.1 and−2.0, respectively.

5. Concluding remarks

In this chapter, we have considered the GDNLS equation (2.1), and constructed a traveling

wave solution (2.19) which is valid for any values of parameters. Motivated by the explicit

form of the solution, we have applied the Painleve test to the GDNLS equation, and shown that

it possesses the Painleve property in strict sense only for the known integrable cases, and the

conditional Painleve property for the cases of Eq. (2.36).

Numerical results imply that the traveling wave solution is stable in the interaction and

behaves like a soliton for smallβ , regardless of the possession of the Painleve property. Re-

markable difference in the behavior of solitary waves between integrable and non-integrable

cases was not observed, except that quantity of ripple generated by the interaction of solitary

waves was small for integrable cases, as was expected.

As for the behavior of solitary waves for largerβ , we could observe the change of shapes

of solitary waves by the interaction. However, it looks that it is still insufficient to conclude that

the solitary waves are not stable. Further theoretical analysis on stability may be necessary.

22

0.00 0.20 0.40 0.60β

−1.0

−0.5

0.0

0.5

FIGURE 2.7. Quantity of position shift (phase shift) per one interaction. Circle:

pulse-1, triangle: pulse-2.

In conclusion, it is expect that the soliton-like behavior of solitary waves for the GDNLS

equation may be a “robust” property. Such behavior may be observed regardless of the value of

parameterβ/α, at least, as far as it is comparably small.

23

0.0 0.2 0.4 0.6 0.8β

0.000

0.010

0.020

0.030

0.040

0.050

[%]

FIGURE 2.8. Quantity of ripple generated after 10 times interactions.β v.s.

ratio of integrated values of ripple to the conserved quantity. Filled circle: the

cases of Eq. (2.28), circle: other cases.

24

CHAPTER 3

An Extension of the Steffensen Iteration and Its Computational

Complexity

1. Introduction

In this chapter, we consider iteration methods for finding a root of a single nonlinear equa-

tion f (x) = 0.

The Newton method is based on a first order approximation of the functionf (x). The

sequence given by it generically converges locally and quadratically to a rootα of f (x). There

have been many attempts to accelerate the Newton method. For example, some methods are

designed based on a higher order approximation (cf. [20]), on a composition of the Newton

iteration [66], on a Pade approximation [64, 16], on a modification off (x) in such a way that

the convergence rate is increased [22, 24], and so on.

The Steffensen method [72] is an iteration method which is applied to a nonlinear equation

of the formx = φ(x). It also has the second order convergence rate, and its iteration function

Φ(x) has no derivative ofφ(x). The Steffensen method can be regarded as a discrete version

of the Newton method. There are so many extensions for the Newton method, however, a few

extension for the Steffensen method. The aim of this chapter is to develop a new iteration

method of the Steffensen type having a higher order convergence rate.

In Section 2, we consider a relationship of the Newton method and the Steffensen method.

In Section 3, we note that the Steffensen iteration functionΦ(x) is congruent with the Aitken

transform [5]. In Section 4, we introduce thek-th Shanks transform [70] which is a natural

extension of the Aitken transform. Whenk = 1, the Shanks transform is reduced to the Aitken

transform. In Section 5, we propose an extension of the Steffensen method in terms of the

k-th Shanks transform. In Section 6, it is proved that the extension has the(k+ 1)-th order

convergence rate provided thatφ ′(α) , 0,±1. Whenφ(α) = 0, the iterated sequence has the

(k+2)2k−1-th order convergence rate. In Section 7, some numerical examples are given which

demonstrate the efficacy of the extended Steffensen iteration. For a special case of the Kepler

equation, it is shown that the numbers of mappings are actually decreased by the extended

Steffensen iteration. In Section 8, we state the remarks of this chapter.

25

2. The Newton method and the Steffensen method

Let us consider the Newton iteration for the equationf (x) = 0. The Newton iteration is

given by

xn+1 = N(xn) := xn− f (xn)f ′(xn)

, n = 0,1, . . . , (3.1)

where the initial approximationx0 is sufficiently close to a rootα . The function f (x) should

be inC2-class on an intervalI such thatα ∈ I . If f ′(α) , 0 andmax|N′(x)| < 1 on I , then the

sequencex0,x1,x2, . . . converges toα quadratically.

To introduce the Steffensen iteration [72], we consider the equationx = φ(x) by setting

φ(x) := x+ f (x) . (3.2)

We prepare the sequencey j generated by the simple iteration

y j+1 = φ(y j) , j = 0,1, . . . . (3.3)

If the sequencey j converges to a numberα, then it follows fromα = φ(α) that f (α) = 0. The

contraction principle guarantees the convergence provided thatmax|φ ′(x)| < 1. Furthermore,

the convergence rate of the sequencey j is linear ifφ ′(α), 0. Let us call suchφ(x) the simple

iteration function.

The Steffensen iterationis an iteration method for finding a root of the nonlinear equation

of the formx = φ(x). There isno derivativein the Steffensen iteration function. Let us define

the recurrence formula

xn+1 = Φ(xn) := xn−

(φ(xn)−xn

)2

φ(φ(xn))−2φ(xn)+xn, n = 0,1, . . . , (3.4)

whereφ(x) is defined by (3.2). HereΦ(x) is the iteration function of the Steffensen iteration

which generates the sequencex0,x1,x2, . . .. If xn → α asn→ ∞, thenα is a root ofx = φ(x).Even if the sequencey j given by the simple iteration (3.3) diverges, the Steffensen iteration

(3.4) may converge toα more faster than does linear order method provided thatφ(x) is in

C1-class,x0 ∈ I and φ ′(α) , 1. Especially, ifφ(x) is in C2-class, the rate isquadratic, or

equivalently, of the second order. The conditionmax|φ ′(x)| < 1 is not necessary in this case

[66, pp. 241–246]. Furthermore, a global convergence theorem is given in [27, pp. 90–95]. See

for an abstract form of the Steffensen iteration [65]. An extension of the Steffensen iteration for

systems of nonlinear equations is proposed in [27, p. 116] and a local convergence theorem is

shown in [63].

26

The Steffensen iteration has its origin in a linear interpolation formula off (x). Let us

briefly review this geometrical feature. A rootα of f (x) = 0 is the intersection point of the

curvey = f (x) and thex-axis inxy-plain (see Figure 3.1). We consider the line through the two

points(a0, f (a0)) and(a1, f (a1)) on the curve. Herea1 is defined by

a1 := φ(a0). (3.5)

The intersection pointα of the line and thex-axis gives an approximation ofα. It follows from

a1−a0 = f (a0) that

α := a0− f (a0)f (a1)− f (a0)

a1−a0

= a0−

(φ(a0)−a0

)2

φ(φ(a0))−2φ(a0)+a0. (3.6)

Thus this approximation formula gives rise to the Steffensen iteration function (3.4). Let us set

h := a1−a0. Taking the limit that the line approaches to the tangential line at(a0, f (a0)), i.e.,

a1→ a0, we derive

α = a0− f (a0)f (a0 +h)− f (a0)

h

→ a0− f (a0)f ′(a0)

as h→ 0. (3.7)

In this limit, α goes to the estimation ofα by the Newton method (3.1). Thus we can regard the

Steffensen iteration asa discrete version of the Newton method. This leads us to believe that an

acceleration of the Steffensen iteration is a meaningful problem.

x

y

f(a0)

f(a1) ααa0 a1

y=f(x)

FIGURE 3.1. Graphical explanation of the Steffensen iteration

27

3. The Steffensen method and the Aitken transform

Let us introduce the Aitken transform [5]. It is a sequence transform to accelerate the

convergence of a given sequencey j. The Aitken transform is given by

y j = y j −(y j+1−y j)2

y j+2−2y j+1 +y j, j = 0,1,2, . . . . (3.8)

If the sequencey j converges to a finite limity∞, then the sequencey j converges to the same

limit y∞ faster thany j. In general (cf. [10, pp. 1–2]), we consider some sequencesSj, Tj,and a sequence transform such thatA : Sj → Tj . If the sequencesSj andTj converge to the

same limitα and satisfy the condition

limj→∞

Tj −αSj −α

= 0, (3.9)

then the sequence transformA is calledsequence convergence accelerator.

The Steffensen iteration functionΦ(xn) is equivalent to the Aitken transform of the three

numbersxn,φ(xn) andφ(φ(xn)). Namely, we have

Φ(xn) = y0 := y0− (y1−y0)2

y2−2y1 +y0, y0 = xn , y1 = φ(xn) , y2 = φ(φ(xn)) (3.10)

for eachn = 0,1, . . .. It should be noted that the sequencey j accelerated by the Aitken

transform is different from the sequencexn generated by the Steffensen iteration (3.4). We

can find thatxn+1 = y0 andxn+2 , y1 in general, even ifxn = y0. In order to use the Aitken

acceleration, we must prepare the whole sequencey j. Moreover, if the convergence rate of

y j is linear, then the convergence rate ofy j is so (cf. [6]). The Aitken acceleration only

guarantees that the sequencey j converges faster thany j does in general. This property is

in sharp contrast to the Steffensen iteration.

4. The Shanks transform and theε-algorithm

Thek-th Shanks transform[70] is a natural extension of the Aitken transform. It is defined

by a ratio of Hankel determinants of2k+1 numbersy j , . . . ,y j+2k by

ek(y j) :=A( j)

k

B( j)k

, j = 0,1,2, . . . . (3.11)

28

Here we define the numeratorA( j)k as a Hankel determinant ofy j , . . . ,y j+2k by

A( j)k :=

k+1︷︸︸︷∣∣∣∣∣∣∣∣∣∣

y j y j+1 · · · y j+k

y j+1 y j+2 · · · y j+k+1...

.... . .

...

y j+k y j+k+1 · · · y j+2k

∣∣∣∣∣∣∣∣∣∣

, (3.12)

and the denominatorB( j)k as a Hankel determinant of∆2y j , . . . ,∆2y j+2k−2 by

B( j)k :=

k︷︸︸︷∣∣∣∣∣∣∣∣∣∣

∆2y j ∆2y j+1 · · · ∆2y j+k−1

∆2y j+1 ∆2y j+2 · · · ∆2y j+k...

.... . .

...

∆2y j+k−1 ∆2y j+k · · · ∆2y j+2k−2

∣∣∣∣∣∣∣∣∣∣

, (3.13)

where∆ is the forward difference operator such that

∆y j := y j+1−y j , ∆2y j := y j+2−2y j+1 +y j . (3.14)

Whenk = 1, the Shanks transform is reduced to the Aitken transformation (3.8). Computation

of determinants usually needs a plenty of multiplications and additions. In order to decrease

the amount of the computations and to avoid the cancellation in the calculation of the Hankel

determinants, we make use oftheε-algorithm [85], [9, pp. 40–51]. The sequenceek(y j)| j =0,1, . . . of the Shanks transform is determined directly by the recurrence relation

ε( j)−1 = 0, ε( j)

0 = y j , j = 0,1,2, . . . , (3.15)

ε( j)i+1 = ε( j+1)

i−1 +1

ε( j+1)i − ε( j)

i

, i = 0,1,2, . . . , j = 0,1,2, . . . , (3.16)

through

ek(y j) = ε( j)2k , j = 0,1, . . . . (3.17)

The amount of computations (3.16) to getek(y j) is only k(2k+2n+1). It should be remarked

that theε-algorithm has a numericalstability.

5. An extension of the Steffensen iteration

The Shanks transform is originally a sequence convergence accelerator for a given sequence.

We apply the Shanks transform to define an iteration function, where the sequencey j is

replaced by that of the simple iterations (3.3). Letx0 be an initial approximation of a rootα of a

29

nonlinear equationx = φ(x). For a fixed natural numberk, we introduce the following iteration

function

xn+1 = Φk(xn) :=Ak(xn)Bk(xn)

, n = 0,1,2, . . . . (3.18)

Here we defineAk(x) andBk(x) as

Ak(x) :=

∣∣∣∣∣∣∣∣∣∣

φ0(x) φ1(x) · · · φk(x)φ1(x) φ2(x) · · · φk+1(x)...

.... . .

...

φk(x) φk+1(x) · · · φ2k(x)

∣∣∣∣∣∣∣∣∣∣

, (3.19)

Bk(x) :=

∣∣∣∣∣∣∣∣∣∣

δ 2φ0(x) δ 2φ1(x) · · · δ 2φk−1(x)δ 2φ1(x) δ 2φ2(x) · · · δ 2φk(x)...

.... . .

...

δ 2φk−1(x) δ 2φk(x) · · · δ 2φ2k−2(x)

∣∣∣∣∣∣∣∣∣∣

. (3.20)

The numberxn+1 becomes a new starting value for the next iteration. Hereφ j(x) andδ 2φ j(x)are compositions of the simple iteration functionφ(x) and their linear combinations defined by

φ0(x) := x, φ j(x) :=

j︷︸︸︷φ(φ(· · ·φ(x) · · ·)) , j = 1,2,3, . . . ,2k, (3.21)

δ 2φ j(x) := φ j+2(x)−2φ j+1(x)+φ j(x) , j = 0,1, . . . ,2k−2, (3.22)

respectively. If a denominator in the formula (3.18) happens to be zero, we setxn+1 = xn.

Especially,Φ1(x) is just the Steffensen iteration function (3.4). Let us call (3.18)the extended

Steffensen iteration.

6. Convergence rate of the extended Steffensen iteration

We now consider the convergence rate of the extended Steffensen iteration (3.18). The main

results in this chapter are as follows.

THEOREM 3.1. If φ(x) is in Ck+1-class andφ ′(α) , 0,±1, then the extended Steffensen

iteration has the(k+ 1)-th order convergence rate. Namely,|xn+1−α| ≤ C|xn−α |k+1 for

some constantC.

Proof. Without loss of generality we can assumeα = 0 whereα is a root ofx = φ(x). We

shall compute the leading term of the Taylor expansion of the iteration functionΦk(x) around

x = 0.

30

Let us perform the following operations to the determinantsAk(x),Bk(x). Setting

cn :=dnφ(0)

dxn , n = 1,2, . . . , (3.23)

we first subtract thei-th row multiplied byc1 from the(i + 1)-th row for i = 1,2, . . .. On the

next step, we subtract thei-th row multiplied byc12 from the(i +1)-th row for i = 2,3, . . .. We

do the similar operations recursively. Then we can express the Hankel determinants (3.19) and

(3.20) as

Ak(x) =

∣∣∣∣∣∣∣∣∣∣

a1,0(x) a1,1(x) · · · a1,k(x)a2,1(x) a2,2(x) · · · a2,k+1(x)

......

. . ....

ak+1,k(x) ak+1,k+1(x) · · · ak+1,2k(x)

∣∣∣∣∣∣∣∣∣∣

, (3.24)

Bk(x) =

∣∣∣∣∣∣∣∣∣∣

b1,0(x) b1,1(x) · · · b1,k−1(x)b2,1(x) b2,2(x) · · · b2,k(x)

......

. . ....

bk,k−1(x) bk,k(x) · · · bk,2k−2(x)

∣∣∣∣∣∣∣∣∣∣

. (3.25)

Here we defineam, j(x) andbm, j(x) as

a1, j(x) := φ j(x) , j = 0,1, . . . ,2k, (3.26)

am+1, j(x) := am, j(x)−c1mam, j−1(x) , m= 1,2, . . . ,k, j = m,m+1, . . . ,2k, (3.27)

b1, j(x) := δ 2φ j(x) , j = 0,1, . . . ,2k−2, (3.28)

bm+1, j(x) := bm, j(x)−c1mbm, j−1(x) , m= 1,2, . . . ,k−1, j = m,m+1, . . . ,2k−2. (3.29)

First we consider the Steffensen case wherek = 1. By the Taylor expansion ofA1(x) we see

a1,0(x) = x, a1,1(x) = c1x+12

c2x2 + · · · , (3.30)

a2,1(x) =12

c2x2 + · · · , a2,2(x) =12

c12c2x2 + · · · . (3.31)

Obviously, we have

A1(x) =12

c1(c1−1)c2x3 + · · · , (3.32)

B1(x) = (c1−1)2x+12(c1

2 +c1−2)c2x2 + · · · . (3.33)

It follows from the conditionc1 , 0,1 thatΦ1(x) = O(x2) asx→ 0. This proves the quadratic

convergence.

31

Next we show thatΦk(x) = O(xk+1) for any natural numberk. The functionsam, j(x) take

the form

am, j(x) = φ j(x)+m−1

∑i=1

β (m)i φ j−i(x) , β (m)

i := (−1)i ∑0<p1<···<pi<m

c1p1+p2+···+pi . (3.34)

This can be checked by using the recurrence relation (3.27). We consider then-th order deriva-

tive of the compositionφ j(x) = φ j−1(φ(x)), which is expressed as

dnφ j(x)dxn =

n

∑r=1

drφ j−1(φ)

dφ r ∑q1+···+qr=nq1≥···≥qr>0

C(q1, . . . ,qr)dq1φ(x)

dxq1· · · d

qr φ(x)dxqr

(3.35)

for n = 1,2, . . .. HereC(q1, . . . ,qr) are unique constants. We define the constantsC(q1,q2, . . .)for qi ∈ 0,1,2, . . ., i = 1,2, . . ., as follows:

(i) C(q1, . . . ,q j ,0) = C(q1, . . . ,q j),(ii) C(. . . ,qi , . . . ,q j , . . .) = 0 if qi < q j ,

(iii) C(1) = 1,

C(q1, . . . ,qr) =r

∑i=1

κ C(q1, . . . ,qi−1,qi−1,qi+1, . . . ,qr) if qr > 0,

whereκ is the number of the non-negative integers having the same values asqi −1 in the set

q1, . . . ,qi−1,qi−1,qi+1, . . . ,qr, namely,

κ := #n = qi−1|n∈ q1, . . . ,qi−1,qi−1,qi+1, . . . ,qr. (3.36)

By use of (3.35) andC(1, . . . ,1) = 1, we writeφ (n)j (0) := dnφ j(0)/dxn as

φ (n)j (0) = c1

nφ (n)j−1(0)+

n−1

∑r=1

γ(n)r φ (r)

j−1(0) , γ(n)r := ∑

q1+···+qr=nq1≥···≥qr>0

C(q1, . . . ,qr)cq1cq2 · · ·cqr . (3.37)

Using (3.34) and (3.37), we see fora(n)m, j(0) := dnam, j(0)/dxn as follows:

a(n)m, j(0) = φ (n)

j (0)+m−1

∑i=1

β (m)i φ (n)

j−i(0)

= c1nφ (n)

j−1(0)+n−1

∑r=1

γ(n)r φ (r)

j−1(0)+m−1

∑i=1

β (m)i

(c1

nφ (n)j−i−1(0)+

n−1

∑r=1

γ(n)r φ (r)

j−i−1(0)

)

= c1n

(φ (n)

j−1(0)+m−1

∑i=1

β (m)i φ (n)

j−1−i(0)

)+

n−1

∑r=1

γ(n)r

(φ (r)

j−1(0)+m−1

∑i=1

β (m)i φ (r)

j−1−i(0)

)

= c1na(n)

m, j−1(0)+n−1

∑r=1

γ(n)r a(r)

m, j−1(0) . (3.38)

32

We insert (3.38) into then-th order derivative of (3.27) to derive

a(n)m+1, j(0) = (c1

n−c1m)a(n)

m, j−1(0)+n−1

∑r=1

γ(n)r a(r)

m, j−1(0) . (3.39)

Assume thatam, j(x) = O(xm), namely,

a(n)m, j(0) = 0, for n < m, (3.40)

a(n)m, j(0) , 0, for n = m. (3.41)

The right hand side of (3.39) is equal to0 for n≤ m. While a(m+1)m+1, j (0) is not equal to0 when

c1 , 0,1. Then it follows that

a(n)m+1, j(0) = 0, for n < m+1, (3.42)

a(n)m+1, j(0) , 0, for n = m+1. (3.43)

This implies thatam+1, j(x) = O(xm+1). By induction we find thatam, j(x) = O(xm) for any

natural numberm. Therefore, the Taylor expansion ofam, j(x) is given by

am, j(x) = a(m)m, j (0)xm+ · · · . (3.44)

On the other hand, we can easily find that

bm, j = am, j+2−2am, j+1 +am, j , m= 1,2, . . . ,k, j = m−1,m, . . . ,2k−2 (3.45)

from the definition (3.29). Then we obtain

b(n)m, j(0) = 0, for n < m, (3.46)

b(n)m, j(0) = (c1

m−1)2a(m)m, j (0) , 0, for n = m (3.47)

from (3.38), (3.44), (3.45) and the conditionc1 ,±1. Hence we have

bm, j(x) = b(m)m, j (0)xm+ · · · (3.48)

for any natural numberm.

Finally we consider the determinantsAk(x) andBk(x). Let Sn be the set of permutations

σ =(

0 1 ··· n−1i0 i1 ··· in−1

)of n-items. By virtue of (3.24), (3.25), (3.44) and (3.48), we see

Ak(x) = ∑σ∈Sk+1

sgnσ ·a1,i0a2,1+i1 · · ·ak+1,k+ik = Lx(k+1)(k+2)/2 + · · · , (3.49)

Bk(x) = ∑σ∈Sk

sgnσ ·b1,i0b2,1+i1 · · ·bk,k−1+ik−1 = M xk(k+1)/2 + · · · . (3.50)

33

Here we define constantL as

L =

∣∣∣∣∣∣∣∣∣∣∣

a(1)1,0(0) a(1)

1,1(0) · · · a(1)1,k(0)

a(2)2,1(0) a(2)

2,2(0) · · · a(2)2,k+1(0)

......

. . ....

a(k+1)k+1,k(0) a(k+1)

k+1,k+1(0) · · · a(k+1)k+1,2k(0)

∣∣∣∣∣∣∣∣∣∣∣

, (3.51)

andM as

M =

∣∣∣∣∣∣∣∣∣∣∣

b(1)1,0(0) b(1)

1,1(0) · · · b(1)1,k−1(0)

b(2)2,1(0) b(2)

2,2(0) · · · b(2)2,k(0)

......

. . ....

b(k)k,k−1(0) b(k)

k,k(0) · · · b(k)k,2k−2(0)

∣∣∣∣∣∣∣∣∣∣∣

. (3.52)

This means thatΦk(x) = O(xk+1). The extended Steffensen iteration defined by thek-th Shanks

transform has the(k+1)-th order convergence rate.

In Theorem 3.1, we use the sequence generated by the simple iteration (3.3) with the itera-

tion function (3.2). In the remaining part of this section, we replace the iteration function (3.2)

by the Newton iteration function (3.1). To this end, let us set the functionφ(x) in (3.18) as

φ(x) := N(x) = x− f (x)/ f ′(x). If f (x) is in C2-class on the intervalI and satisfyf ′(α) , 0 and

f ′′(α) , 0, then the functionφ(x) satisfiesφ ′(α) = 0 andφ ′′(α) , 0 and the Newton iteration

y j+1 = φ(y j) locally converges toα quadratically. We have

THEOREM 3.2. If φ(x) is in C(k+2)2k−1-class andφ ′(α) = 0, φ ′′(α) , 0, then the extended

Steffensen iteration has the(k+2)2k−1-th order convergence rate. Namely,|xn+1−α| ≤C|xn−α|(k+2)2k−1

for some constantC.

Proof. We restrict ourselves to the case whereα = 0, for simplicity. Along the line which

is similar to Theorem 3.1, we shall compute the Taylor coefficientsφ (n)j (0) of φ j(x). From

(3.37) and the conditionsc1 = 0, c2 , 0, it is turned out that

φ (n)j (0) = 0, for n < 2 j −1, (3.53)

φ (n)j (0) , 0, for n = 2 j . (3.54)

Then we findφ j(x) = O(x2 j) andδ 2φ j(x) = O(x2 j

). We consider the Hankel determinant

Ak(x) = ∑σ∈Sk+1

sgnσ ·φi0φ1+i1φ2+i2 · · ·φk+ik . (3.55)

34

The leading term ofAk(x) is given by the termφi0φ1+i1 · · ·φk+ik = O(x2i0+···+2k+ik) which has

the minimal degree inx. The degree becomes minimal wheni0 = k, i1 = k−1, i2 = k−2, . . .,

ik=0. It follows thatAk(x) = O(x(k+1)2k). Similarly, Bk(x) = O(xk2k−1

). Consequently, we see

Φk(x) = O(x(k+2)2k−1) which completes the proof of Theorem 3.2.

In the book of Ostrowski [66, p. 252] a composition of the Newton iterations is formulated

which has third-order convergence. The iteration in Theorem 3.2 withk = 1 provides third-

order convergence. The extended Steffensen iteration in this case is also a composition of the

Newton iterations, however, it is rather different from that in [66].

7. Numerical examples and computational complexity

In this section we present explicit examples to demonstrate how the extended Steffensen

iteration acts. The computational complexity is also discussed.

All results of the numerical experiments are computed on the Intel Pentium Pro Processor

200 MHz. In Example 1 and 2, we examine the new iteration methods by use of the Mathe-

matica version 3 (Wolfram Research, Inc.). In Example 3, we program them by the GNU C

compiler version 2.7.2.

Example 1. The nonlinear equation to be solved is

f (x) = exp(−x)−x = 0, (3.56)

which has the unique solutionα = 0.56714329040978104129· · · . In order to apply Theorem

3.1, we set the iteration functionφ(x) as

φ(x) = exp(−x). (3.57)

It should be noted thatφ ′(α) , 0,±1 and φ(x) satisfies the condition of Theorem 3.1. We

compare several iteration methods. They are the simple iteration (3.3), the Steffensen itera-

tion (3.4), and the extended Steffensen iteration (3.18) withk = 2,3,4. We choose the initial

approximation asx0 = 0, and generate the sequencexn until the condition

| f (xn∗)|< 10−r , r = 1000 (3.58)

is satisfied. Thenxn∗ gives an approximation of the solutionα. We compute the sequences in

the multi precision arithmetic. In Figure 3.2, the quantitylog10| f (xn)| is illustrated to estimate

the error. In Table 3.1, we give the numbern∗ of iterations and an estimation of the convergence

35

rate,

log10

∣∣∣∣xn∗−1−xn∗

xn∗−2−xn∗

∣∣∣∣

log10

∣∣∣∣xn∗−2−xn∗

xn∗−3−xn∗

∣∣∣∣, (3.59)

by using four numbersxn∗−3,xn∗−2,xn∗−1 andxn∗.

It is shown that the iteration numbersn∗ crucially depend on the iteration methods. On the

convergence rate in Table 3.1, the estimated values are very close to the theoretical values for

all iterations.

Example 2. Let us consider the same equationf (x) = exp(−x)−x = 0 as in Example 1.

We here replace the iteration functionφ(x) by the Newton iteration function

φ(x) = x+exp(−x)−xexp(−x)+1

. (3.60)

Obviously,φ ′(α) = 0, φ ′′(α) , 0. Namely,φ(x) holds the condition in Theorem 3.2. Set the

initial approximation asx0 = 0. The sequences are computed in the multi precision arithmetic.

In Figure 3.3 and Table 3.2, the Newton method (3.1) and the extended Steffensen iteration

(3.18) withk = 1,2,3,4 are illustrated. The estimated convergence rates seem to be good ap-

proximations of the theoretical rates.

Example 3. To discuss the computational complexity and the convergence property we

solve the Kepler equation

f (x) := x− l −esin(x) = 0 (3.61)

TABLE 3.1. Number of iterations and convergence rate. (Example 1)

numbern∗ of iterationsconvergence rate

numerical theoretical

simple iteration 4059 1.00† 1

Steffensen iteration 10 2.00 2

extended Steffensen iteration,k = 2 7 3.00 3



† This value is obtained byxi ,xi+1,xi+2 andxn∗ for i = 10,11, . . . ,4045. For i > 4045, the estimation of

the convergence rate is quite different from 1.00.

36

0 5 10 15 20itr. # n

−1000

−800

−600

−400

−200

0

log 10

|f(x n)

|

FIGURE 3.2. A comparison oflog10| f (xn)| of several iteration methods when

φ ′(α) , 0,±1. (Example 1) Solid line: simple iteration. Dashed line: Stef-

fensen iteration. Circles, squares and triangles denote the extended Steffensen

iteration fork = 2,3, and4, respectively.

0 5 10itr. # n

−1000

−800

−600

−400

−200

0

log 10

|f(x n)

|

FIGURE 3.3. A comparison oflog10| f (xn)| of several iteration methods when

φ ′(α) = 0, φ ′′(α) , 0. (Example 2) Dashed line: Newton method. Pluses,

circles, squares and triangles denote the extended Steffensen iteration fork =1,2,3, and4, respectively.

37

for variousl ande, by using the simple iteration, the Newton method, the Steffensen iteration

and the extended Steffensen iteration withk = 2. The Kepler equation appears in orbit deter-

mination in celestial mechanics andx, l ande are the eccentric anomaly, the mean anomaly

and the eccentricity, respectively. We solve the Kepler equation forx, where the remaining pa-

rametersl ande are fixed such that0≤ l ≤ π, 0 < e≤ 1. Let x0 = l be the initial value. Let

us setφ(x) := l +esin(x) and insertφ(x) into the iteration functions of the Steffensen and the

extended Steffensen iterations. We use| f (xn∗)|< 10−13 as the stopping criterion in the double

precision arithmetic.

We first show the convergence property of the iterations. The simple iteration always con-

verges for any pair ofl ande. The marks in Figures 3.4, 3.5, and 3.6, indicate the pairs(l ,e)for which the iterations do not converge. The mesh sizes ofl ande in the figures are0.01π/180

and0.001, respectively. We see that the Steffensen type iterations converges in more cases than

the Newton method. There are some parameters for which the Steffensen iteration converge but

TABLE 3.2. Number of iterations and convergence rate. (Example 2)

numbern∗ of iterationsconvergence rate

numerical theoretical

Newton method 11 2.00 2




extended Steffensen iteration,k = 4 2 —‡ 48

‡ Sincexn∗−3 dose not exist, it is impossible to estimate the convergence rate.

TABLE 3.3. Number of iterations and total numbers of mappings. (Example 3)

numbern∗ of iterations total numbers of mappings

average maximal l = 18π180 average maximal l = 18π

180

e= 0.95 e= 0.95

simple iteration 45.94 2903 33 45.94 2903 33

Newton method 10.18 886 30 20.36 1772 60

Steffensen iteration 3.88 30 30 7.76 60 60

extended Steffensen 3.87 632 7 15.48 2528 28

iteration,k = 2

38

the extended Steffensen iteration does not. The ratios of the number of all grid points to that of

the marks in Figures 3.4, 3.5, and 3.6 are 0.06400% (Newton method), 0.02732% (Steffensen

iteration) and 0.03536% (the extended Steffensen iterations), respectively.

0 30 60 90 120 150 180mean anomaly l [deg]

0.0

0.2

0.4

0.6

0.8

1.0ec

cent

rici

ty e

FIGURE 3.4. The parameters(l ,e) for which the Newton iterations do not con-

verge. (Example 3)


0.0

0.2

0.4

0.6

0.8

1.0

ecce

ntri

city

e

FIGURE 3.5. The parameters(l ,e) for which the Steffensen iterations do not

converge. (Example 3)

39

Next, we illustrate the computational complexity with Table 3.3. We solve the Kepler

equation for all parameters(l ,e) such thatl = i π/180, i = 0,1, . . . ,180 ande = 0.01 j, j =1,2, . . . ,100. The maximal and averaged numbers of iterations of each iteration method are

shown in Table 3.3. The amount of computations of theε-algorithm in the extended Steffensen

iteration is negligible as compared with that of the mappingφ . Thusthe total numbers of map-

pingsare essential as well as the numbers of iterations in order to estimate the computational

complexity. The simple iteration, the Newton method, the Steffensen iteration and the extended

Steffensen iteration (k = 2), respectively, needs 1, 2, 2 and 4 mappings in one iteration. The to-

tal numbers of mappings are also shown in Table 3.3. The averaged and maximal total numbers

of mappings of the Steffensen iteration is less than those of any other methods. However, the

Steffensen iteration is the worst whenl = 18π/180, e= 0.95. While the extended Steffensen

iteration works well. For these special parameters, the extended Steffensen iteration is superior

than other iterations.


In this chapter, we consider an extension of the Steffensen iteration in terms of the Shanks

transform. The resulting iteration method does not need any derivatives and has a higher or-

der convergence rate. Ifφ(y j) converges linearly, then the sequenceΦk(xn) defined by

using thek-th Shanks transform has the(k+ 1)-th order convergence rate (see Theorem 3.1).


0.0

0.2

0.4

0.6

0.8

1.0

ecce

ntri

city

e

FIGURE 3.6. The parameters(l ,e) for which the extended Steffensen iterations

for k = 2 do not converge. (Example 3)

40

HereΦ1(x) is just the Steffensen iteration function. On the other hand, ifφ(y j) converges

quadratically, like the Newton sequence, then the iterated sequenceΦk(xn) has remarkably

the (k+ 2)2k−1-th order convergence rate (see Theorem 3.2). These theoretical convergence

rates can be found in numerical examples (Examples 1, 2).

For the implementation of the extended Steffensen iteration, the stableε-algorithm is espe-

cially useful to decrease the amount of computations in the calculation of Hankel determinants.

Consequently, the numbers of mappings take a major part of the computational complexity. It

is shown (Example 3) that the extended Steffensen iteration withk = 2 has the minimal num-

bers of mappings in a special case of the Kepler equation. Moreover, the extended Steffensen

iteration converges for more cases of parameters than the Newton method.

After the completion of this research the authors are told the references [10], [48] by Pro-

fessor N. Osada, which considers a generalized Steffensen iteration without any discussion on

computational complexity. The idea in [48] is essentially the same as that in this thesis, however,

there is no explicit numerical examples and no comparison to other iteration methods.

41

CHAPTER 4

Determinantal Solutions for Solvable Chaotic Systems and Iteration

Methods Having Higher Order Convergence Rates

1. Introduction

The singularity confinement (SC) is a useful integrability criterion for discrete nonlinear

dynamical systems [25]. The discrete Painleve equations and many discrete soliton equations

pass the SC test. However the SC test is not sufficient to identify integrability. In the literature

[28], Hietarinta and Viallet presented a discrete dynamical system which passes the SC test but

possesses a numerically chaotic property. Then they proposed a more sensitive integrability test

[28, 8] using the algebraic entropy. The algebraic entropy is defined by the logarithmic average

of a growth of degrees of iterations. Both test are similar to each, and the algebraic entropy test

is a more precise criterion than the SC test.

Many of good numerical algorithms are deeply connected to the nonlinear integrable sys-

tems. For example, the recurrence relation of the qd-algorithm, which is used for calculating

a continued fraction, is equivalent to the discrete time Toda equation. And the recurrence rela-

tion of theε-algorithm [85], which is a sequence convergence accelerator, is equivalent to the

discrete potential KdV equation. From these results, one may conjecture that good numerical

algorithms can be regared as integrable dynamical systems. Indeed, many of linearly convergent

algorithms such as eigenvalue algorithms and sequence accelerators pass the SC type criteria

(cf. [68]), and they are proved to be equivalent to soliton equations. However, the algorithms

having higher order convergence rates, which give irreversible dynamical systems, do not pass

the SC type criteria. The techniques in the nonlinear integrable systems cannot be directly

adapted to them.

The arithmetic-harmonic mean (AHM) algorithm [62] is an irreversible system having an

explicit solution, however does not pass the SC type criteria. According to the setting of initial

conditions, it behaves as an algorithm having the second order convergence rate, or as a solvable

chaotic system. In this chapter, we investigate such discrete dynamical systems and obtain their

determinantal solutions. We deal with the Ulam-von Neumann (UvN) system [77] which is

a solvable chaotic system, and with the discrete dynamical systems derived from the Newton

method, an extension of the Newton method, the Steffensen method [72], and the extended

42

Steffensen method proposed in Chapter 3, which are iteration methods having higher order

convergence rates.

In Section 2, we show the trigonometric solutions for the AHM algorithm and the UvN

system in terms of addition formulas. Moreover we show the hierarchy of the UvN system.

The AHM algorithm is equivalent to the Newton method for a quadratic equation. In Section

3, we introduce the Newton method and the Nourein method [64, 16] which is an extension of

the Newton method. Applying these methods to a quadratic equation, we present the hierarchy

of the Newton type iterations. In Section 4, we give addition formulas of the determinants of

certain tridiagonal matrices. In Section 5, we show determinantal solutions for the discrete Ric-

cati equation. In Section 6, we obtain determinantal solutions for the hierarchy of the Newton

type iterations. In Section 7, determinantal solutions for the hierarchy of the UvN system are

derived. In Section 8, we obtain determinantal solutions for the hierarchy of the Steffensen type

iterations. In Section 9, we give some remarks.

2. Trigonometric solutions for solvable chaos systems

In this section, we introduce solvable chaotic systems which have trigonometric solutions.

We shall show that these solutions are obtained in terms of some addition formulas.

Firstly, we consider the iteration

an+1 =an +bn

2, bn+1 =

2anbn

an +bn, n = 0,1,2, . . . , (4.1)

which is called the arithmetic-harmonic mean (AHM) algorithm [62]. The AHM algorithm has

the following solutions. For the casea0 > b0 > 0, we have

an = N1coth(2nσ1) , bn = N1 tanh(2nσ1) . (4.2)

For the casea0 > 0, b0 < 0, we have

an = N2cot(2nσ2) , bn =−N2 tan(2nσ2) . (4.3)

Here the positive constantsN1, N2, σ1 andσ2 are uniquely determined by the initial valuesa0

andb0. The solutions (4.2) and (4.3) are derived from the double angle formulas ofcoth(x) and

cot(x),

coth(2x) =coth(x)+ tanh(x)

2, tanh(2x) =

2 coth(x) tanh(x)coth(x)+ tanh(x)

, (4.4)

cot(2x) =cot(x)− tan(x)

2, tan(2x) =

2 cot(x) tan(x)cot(x)− tan(x)

, (4.5)

43

respectively. The AHM algorithm has the conserved quantityI = anbn, which can be easily

checked by (4.1). ThusI = a0b0. Using the conserved quantityI , we introduce the variableun

such thatun = an = I/bn. Then we have the discrete dynamical system

un+1 =12

(un +

Iun

). (4.6)

The system (4.6) can be also derived by applying the Newton method to the quadratic equation

f (z) = z2− I = 0. The behaviors ofun are illustrated in Figures 4.1 and 4.2. When the

caseI = a0b0 > 0, the sequenceun quadratically converges to the positive root ofI (see Figure

0 2 4 6 8 10

n

0.4

0.5

0.6

0.7

0.8

0.9

1

u n

FIGURE 4.1. Behavior of the Newton method (4.6) for the caseI = a0b0 > 0.

0 20 40 60 80 100

n

−20

0

20

40

60

u n

FIGURE 4.2. Behavior of the Newton method (4.6) for the caseI = a0b0 < 0.

44

4.1). When the caseI = a0b0 < 0, it behaves as a solvable chaotic system (see Figure 4.2). Its

invariant measure isµ(dx) = dx/(π(1+x2)), and its Lyapunov exponent islog2 (cf. [79]).

Next, we consider the solvable logistic map, or the Ulam-von Neumann (UvN) system [77],

0 < u0 < 1, un+1 = 4un(1−un) , n = 0,1,2, . . . . (4.7)

A solution for (4.7) is obtained by

un = sin2(2nσ3) , (4.8)

which is derived from the double angle formula ofsin2(x),

sin2(2x) = 4sin2(x)(1−sin2(x)) . (4.9)

Here the constantσ3 is determined by the initial valueu0. The invariant measure of the UvN

system isµ(dx) = dx/(π√

x(1−x)), and the Lyapunov exponent of it islog2 (cf. [79]). By

virtue of then-tuple angle formulas of trigonometric functions, the higher order systems of the

UvN system are given by

u(2)n+1 = 4u(2)

n (1−u(2)n ) , (4.10)

u(3)n+1 = u(3)

n (3−4u(3)n )2 , (4.11)

u(4)n+1 = 16u(4)

n (1−u(4)n )(1−2u(4)

n )2 , (4.12)

u(5)n+1 = u(5)

n (5−4u(5)n (5−4u(5)

n ))2 , (4.13)

and so on (cf. [80]). The superscriptsmof u(m)n denote the order of the hierarchy. Their invariant

measures are allµ(dx) = dx/(π√

x(1−x)), and their Lyapunov exponents are respectively

logm for m= 3,4,5, . . .. Another generalization of the UvN system having Jacobi or Weierstrass

elliptic function solution is discussed in [78].

An aim of this chapter is to obtain determinantal solutions for the discrete dynamical sys-

tems (4.6), (4.7) and their hierarchies. The hierarchy of (4.6) is introduced in Section 3, and the

hierarchy of (4.7) already appear above.

3. The Newton method and the Nourein method

In this section, we introduce the Newton method (cf. [13]) and an extension of the Newton

method for finding a root of an equationf (z) = 0. Furthermore we present a hierarchy of

discrete dynamical systems given by the Newton type iterations.

45

The Newton method is given by

un+1 = N(un) , n = 0,1,2, . . . , (4.14)

N(z) = z− f (z)f ′(z)

. (4.15)

Here the prime denotesf ′(z) = d f(x)/dz. The Nourein method [64, 16], which is an extension

of the Newton method based on the Pade approximation, is given by

un+1 = Np(un) , n = 0,1,2, . . . , (4.16)

Np(z) = z− f (z)Hp(z)

Hp+1(z), (4.17)

whereHp(z) are defined by

H0(z) = 1, Hp(z) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

c1 c0 0 0 · · · 0

c2 c1 c0 0 · · · 0...

......

... · · · ...

cp−1 cp−2 cp−3 cp−4 · · · c0

cp cp−1 cp−2 cp−3 · · · c1

∣∣∣∣∣∣∣∣∣∣∣∣∣

, p = 1,2,3, . . . , (4.18)

andc j(z) denote

c j(z) =1j!

d j f (z)dzj , j = 0,1,2, . . . . (4.19)

The convergence rate of the Nourein method is of orderp+2. Whenp = 0, 1, and2, then the

Nourein method (4.16) is reduced to the Newton method, the Halley method (cf. [19, pp. 220–

221], [64]), and the Kiss method (cf. [64]), respectively.

Applying the Newton method (4.14) and the Nourein method (4.16) to the quadratic equa-

tion

f (z) := z2 +2bz+c = 0, (4.20)

we obtain the following discrete dynamical systems forp = 0,1,2, . . .,

u(2)n+1 =

(u(2)n )2−c

2u(2)n +2b

, (4.21)

u(3)n+1 =

(u(3)n )3−3cu(3)

n −2bc

3(u(3)n )2 +6bu(3)

n +(4b2−c), (4.22)

u(4)n+1 =

(u(4)n )4−6c(u(4)

n )2−8bcu(4)n − (4b2−c)c

4(u(4)n )3 +12b(u(4)

n )2 +4(4b2−c)u(4)n +4b(2b2−c)

, (4.23)

46

and so on. The superscriptsm := p+2 of u(m)n denote the order of the hierarchy. In Section 6,

we shall obtain determinantal solutions for the hierarchy of the discrete systems (4.21)–(4.23).

4. Addition formula for tridiagonal determinant

In order to get solutions for the discrete dynamical systems corresponding to iteration meth-

ods and solvable chaotic systems, we derive an addition formula for tridiagonal determinants,

which is an extension of addition formula for trigonometric function.

In this section, we present four lemmas for determinants. Let us consider the sequence of

determinants of tridiagonal matrices,

τn :=

n︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣

α β

1 α . . .. . . . . . β

1 α

∣∣∣∣∣∣∣∣∣∣∣

, n = 1,2,3, . . . , (4.24)

whereα andβ are arbitrary complex constants. We setτ−1 := 0 andτ0 := 1. It should be noted

thatτn is a monic polynomial ofα of degreen. We can prove the following elementary lemmas.

LEMMA 4.1 (Three-term recurrence relation).

τ−1 = 0, τ0 = 1, τn+1 = α τn−β τn−1 , n = 0,1,2, . . . . (4.25)

Proof. In terms of the expansion of the determinantτn+1 with respect to the last row, we

derive (4.25).

Here let us assume thatβ is a real positive constant. Setting

x :=α

2√

β, T0(x) := 1, Tn(x) :=

τn

2√

β n− τn−2

2√

β n−2, n = 1,2, . . . , (4.26)

we obtain the recurrence relation

T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x)−Tn−1(x) , n = 1,2, . . . , (4.27)

from (4.25). The functionsTn(x) are the Chebyshev polynomials of the first kind. The Cheby-

shev polynomials can be also expressed as

Tn(x) := cos(n arccos(x)) . (4.28)

Thus the determinantsτn can be related to the trigonometric functions.

47

LEMMA 4.2 (Addition formula).

τn+m = τnτm−β τn−1τm−1 , n,m= 0,1, . . . . (4.29)

Proof. Formula (4.29) is a consequence of the Laplace expansion (cf. [71]) for the deter-

minantτn+m with respect to the firstn rows. We give an alternative proof here. Let us assume

thatτn , 0. From a determinant partitioning formula for block matrices, it follows that

τn+m = τn

∣∣∣∣∣∣∣∣∣∣∣∣

α β

1 α . . .. . . . . . β

1 α

−

1

α β

1 α . . .. . . . . . β

1 α

−1

β

∣∣∣∣∣∣∣∣∣∣∣∣

m.

(4.30)

We then have

τn+m = τn

∣∣∣∣∣∣∣∣∣∣∣

α−βτn−1/τn β

1 α . . .. . . . . . β

1 α

∣∣∣∣∣∣∣∣∣∣∣

. (4.31)

Expanding the first row, we obtain

τn+m = τn(ατm−1−βτm−2)−βτn−1τm−1 . (4.32)

Using Lemma 4.1, we derive (4.29). We have proved Lemma 4.2.

LEMMA 4.3 (Linear-bilinear identity).

2τmτn− τm−1τn+1− τm+1τn−1

= β n(2τm−n−α τm−n−1) , m≥ n, n = 0,1,2, . . . . (4.33)

Proof. From Lemma 4.2, it follows that

2τmτn− τm−1τn+1− τm+1τn−1 = β (2τm−1τn−1− τm−2τn− τmτn−2) . (4.34)

Note that the indices of the right hand side of this relation are decreased by1 rather than those

of the left hand side. Calculating this relation recursively, we have

2τmτn− τm−1τn+1− τm+1τn−1 = β n(2τm−nτ0− τm−n−1τ1− τm−n+1τ−1) . (4.35)

Fromτ−1 = 0, τ0 = 0 andτ1 = α , this relation becomes to (4.33). We have proved Lemma 4.3.

48

LEMMA 4.4 (Differential relation). If β is independent ofα , then

∂τn+2

∂α−β

∂τn

∂α= (n+2)τn+1 , n = 0,1, . . . . (4.36)

If α andβ depend on the same parametert then

∂τn+2

∂ t−β

∂τn

∂ t= (n+2)

∂α∂ t

τn+1− (n+1)∂β∂ t

τn , n = 0,1, . . . . (4.37)

Proof. A partial differentiation with respectα leads to

∂τn+2

∂α=

n+1

∑j=0

τ jτn+1− j . (4.38)

From Lemma 4.2, it follows that

∂τn+2

∂α= (n+2)τn+1 +β

n

∑j=1

τ j−1τn− j = (n+2)τn+1 +βn−1

∑j=0

τ jτn−1− j . (4.39)

From (4.38), we obtain

∂τn+2

∂α= (n+2)τn+1 +β

∂τn

∂α. (4.40)

Thus we have proved (4.36). The relation (4.37) can be proved by a similar line of thought.

5. Determinantal solution for the discrete Riccati equation

In order to get solutions for the discrete dynamical systems corresponding to the Newton

type iterations and the Steffensen type iterations, we give the determinantal solution for the

discrete Riccati equation.

Let us consider the discrete Riccati equation

xn+1 =axn +bcxn +d

, n = 0,1,2, . . . , (4.41)

wherexn is a complex variable, anda, b, c andd are complex constants. When we set the

parameters as

xn = X(t) , t = nδ , a = 1+Bδ , b = Cδ , c =−Aδ , d = 1−Bδ , (4.42)

and take the limit asδ → 0, then we have the differential Riccati equation

dX(t)dt

= AX(t)2 +2BX(t)+C (4.43)

with the constant coefficientsA, B andC.

A determinantal solution for Eq. (4.41) is obtained by

xn =x0τn− (x0d−b)τn−1

τn− (a−x0c)τn−1, n = 0,1,2, . . . . (4.44)

49

Hereτn is the determinant of the degreen defined by

τ−1 = 0, τ0 = 1, τn =

n︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

α β1 α β

. . .. . .

. . .

1 α β1 α

∣∣∣∣∣∣∣∣∣∣∣∣∣

, n = 1,2,3, . . . , (4.45)

whereα andβ denoteα = a+d, β = ad−bc.

From Lemma 4.1, the determinantsτn satisfy the linear difference equation

τ−1 = 0, τ0 = 1, τn+1− (a+d)τn +(ad−bc)τn−1 = 0, n = 0,1,2, . . . . (4.46)

Substituting (4.44) into (4.41) using (4.46), we check that (4.44) gives a solution.

6. Determinantal solutions for hierarchy of the Newton iteration

In this section, we obtain determinantal solutions for the hierarchy of the Newton type

iterations (4.21)–(4.23). The hierarchy is derived by applying the Newton type methods (4.14),

(4.16) to the quadratic equationf (z) = z2 +2bz+c.

6.1. Determinantal solutions.We begin to consider the following discrete Riccati equa-

tion

vn+1 =v0vn−c

vn +(v0 +2b), n = 0,1,2, . . . , (4.47)

whereb, c andv0 are arbitrary complex values. Note that the initial valuev0 is included in the

coefficients of the recurrence relation. From the determinantal solution for the discrete Riccati

equation in Section 5, we obtain a solution for (4.47)

vn = v0−BFn−1

Fn, n = 0,1,2, . . . . (4.48)

HereA andB denoteA = f ′(v0), B = f (v0), andFn are the determinants defined by

F−1 = 0, F0 = 1, Fn =

n︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

A B

1 A B. . . . . . . . .

1 A B

1 A

∣∣∣∣∣∣∣∣∣∣∣∣∣

, n = 1,2,3, . . . . (4.49)

Next, we consider addition formulas forvn, which are resulted from the following theorem.

50

THEOREM4.1 (Addition formula). The solutionvn (4.48)for Eq.(4.47)satisfies the relation

v(m+1)n−1 =vmn−1vn−1−c

vmn−1 +vn−1 +2b, m,n = 1,2,3, . . . . (4.50)

This relation gives them-tuple addition formulas

vmn−1 = Nm−2(vn−1) , m= 2,3,4, . . . , n = 0,1,2, . . . , (4.51)

where we define functionsNp(z) by

Np(z) := z− f (z)Hp(z)

Hp+1(z), p = 0,1,2, . . . , (4.52)

H0(z) := 1, Hp(z) :=

p︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

f ′(z) f (z)1 f ′(z) f (z)

. . . . . . . . .

1 f ′(z) f (z)1 f ′(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣

. (4.53)

Proof. First we shall prove the relation (4.50). From Lemmas 4.1 and 4.2, the determinants

F(m+1)n−2 andF(m+1)n−1 are given by

F(m+1)n−2 = Fmn−1Fn−1−BFmn−2Fn−2 , (4.54)

F(m+1)n−1 = AFmn−1Fn−1−BFmn−1Fn−2−BFmn−2Fn−1 . (4.55)

Inserting (4.54) and (4.55) into

v(m+1)n−1 = v0−BF(m+1)n−2

F(m+1)n−1, (4.56)

we have

v(m+1)n−1 = v0−BFmn−1Fn−1−BFmn−2Fn−2

AFmn−1Fn−1−BFmn−2Fn−1−BFmn−1Fn−2. (4.57)

Rearranging (4.57), we obtain

v(m+1)n−1 =

(v0−B

Fmn−2

Fmn−1

)(v0−B

Fn−2

Fn−1

)−c

(v0−B

Fmn−2

Fmn−1

)+

(v0−B

Fn−2

Fn−1

)+2b

. (4.58)

The relation (4.58) leads to the proof of (4.50).

Next we shall prove the addition formulas (4.51). Whenm= 2, the formula (4.51) can be

easily shown by using (4.50). Let us assume thatvmn−1 satisfy (4.51) for a certainm≥ 2. Then

51

we shall check thatv(m+1)n−1 satisfy the relation (4.51). From the assumption, we rewrite (4.50)

as

v(m+1)n−1 =Nm−2(vn−1)vn−1−c

Nm−2(vn−1)+vn−1 +2b. (4.59)

From (4.52) it follows that

v(m+1)n−1 = vn−1− f (vn−1)Hm−1(vn−1)(2vn−1 +2b)Hm−1(vn−1)− f (vn−1)Hm−2(vn−1)

. (4.60)

SinceHp(z) satisfy

Hp(z) = f ′(z)Hp−1(z)− f (z)Hp−2(z) , (4.61)

we have

v(m+1)n−1 = vn−1− f (vn−1)Hm−1(vn−1)Hm(vn−1)

= Nm−1(vn−1) . (4.62)

By induction, the addition formulas (4.51) are proved.

Finally, we introduce the variablesu(m)n defined by

u(m)n := vmn−1 , m= 2,3,4, . . . , n = 0,1,2, . . . . (4.63)

Thus let us consider the map

u(m)0 = v0 7→ u(m)

1 = vm−1 7→ u(m)2 = vm2−1 7→ u(m)

3 = vm3−1 7→ · · · . (4.64)

By virtue of them-tuple addition formulas (4.51), we thus obtain the hierarchy of the discrete

dynamical systems

u(2)n+1 =

(u(2)n )2−c

2u(2)n +2b

, (4.65)

u(3)n+1 =

(u(3)n )3−3cu(3)

n −2bc

3(u(3)n )2 +6bu(3)

n +(4b2−c), (4.66)

u(4)n+1 =

(u(4)n )4−6c(u(4)

n )2−8bcu(4)n − (4b2−c)c

4(u(4)n )3 +12b(u(4)

n )2 +4(4b2−c)u(4)n +4b2(2b2−c)

, (4.67)

and so on. These discrete systems are the same as the Newton type iterations (4.21)–(4.23).

Therefore we obtain the determinantal solutions for the hierarchy of the Newton iterations by

u(m)n = u(m)

0 −BFmn−2

Fmn−1, m= 2,3,4, . . . , n = 0,1,2, . . . , (4.68)

from (4.48), (4.49), (4.63) andA = f ′(u(m)0 ), B = f (u(m)

0 ).52

It is to be remarked that the determinantal solution (4.68) is also expressed as the continued

fraction

u(m)n = u(m)

0 − B

A− B

A· · ·− B

A.

︸︷︷︸mn−1

(4.69)

6.2. Other solutions. In the previous subsection, we have constructed the determinantal

solutions (4.68) in terms of only four arithmetic operations. Here we ease this restriction. Let

us allow to use the operation of square root. Then solutions of other type are obtained as follows,

u(m)n = u(m)

0 − r2 r1mn− r1 r2

mn

r1mn− r2

mn , n = 0,1,2, . . . . (4.70)

Herer1 andr2 are the roots of the characteristic equation

x2−Ax+B = 0, A = f ′(u(m)0 ) , B = f (u(m)

0 ) , (4.71)

which is given by the three-term recurrence relation ofFn. When we use the rootsλ1, λ2 of

f (z) = 0, then we have

u(m)n =

λ2(u(m)0 −λ1)mn−λ1(u(m)

0 −λ2)mn

(u(m)0 −λ1)mn− (u(m)

0 −λ2)mn, n = 0,1,2, . . . . (4.72)

The solution (4.72) is also expressed as

u(m)n = (ψ−1Rm

nψ)(u(m)0 ) , (4.73)

where we define the functionsR(z), ψ(z) as

Rm(z) := zm, ψ(z) :=z−λ1

z−λ2. (4.74)

This result implies that the mapNm−2 is conjugate with the mapRm, namely

Nm−2 = ψ−1Rmψ . (4.75)

The relation (4.75) yields the Julia set of the mapNm−2 by

J(Nm−2) = w|w = ψ−1(z), |z|2 = 1,z∈ C . (4.76)

The relation (4.75) withm= 2 was originally found by Cayley in 1879 (cf. [67]) for the Newton

method.

53

7. Determinantal solutions for hierarchy of the Ulam-von Neumann system

7.1. Determinantal solutions.We begin to consider the following linear difference equa-

tion

v−1 = 2, v0 = A, vn+1−Avn +Bvn−1 = 0, n = 0,1,2, . . . , (4.77)

whereA, B are arbitrary complex constants. A determinantal solution for Eq. (4.77) is obtained

by

vn = Fn+1−BFn−1 , n = 0,1,2, . . . , (4.78)

whereFn is the determinant of the degreen defined by

F−1 = 0, F0 = 1, Fn =

n︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

A B

1 A B. . .

. . .. . .

1 A B

1 A

∣∣∣∣∣∣∣∣∣∣∣∣∣

, n = 1,2,3, . . . . (4.79)

Next we consider addition formulas forvn, which are given by the following theorem.

THEOREM 4.2 (Addition formula). The solutionvn for Eq. (4.77)satisfies the relation

v(m+1)n−1 = vmn−1vn−1−Bnv(m−1)n−1 , m= 1,2,3, . . . , n = 1,2,3, . . . . (4.80)

This relation gives them-tuple addition formulas

vmn−1 = Gm(vn−1)−BnGm−2(vn−1) , m= 2,3,4, . . . , n = 1,2,3, . . . , (4.81)

whereGm(z) are defined by

G0(z) = 1, Gm(z) =

m︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

z Bn

1 z Bn

. . .. . .

. . .

1 z Bn

1 z

∣∣∣∣∣∣∣∣∣∣∣∣∣

, m= 1,2,3, . . . . (4.82)

54

Proof. First we shall prove the relation (4.80). From (4.78) and Lemma 4.2, it follows that

v(m+1)n−1 = Fmn+n−BF(mn−1)+(n−1) (4.83)

= (FmnFn−BFmn−1Fn−1)−B(Fmn−1Fn−1−BFmn−2Fn−2)

= (Fmn−BFmn−2)(Fn−BFn−2)−B(2Fmn−1Fn−1−Fmn−2Fn−FmnFn−2) .

From (4.78) and Lemma 4.3, we have

v(m+1)n−1 = vmn−1vn−1−Bn(2Fmn−n−AFmn−n−1) . (4.84)

From (4.78) and Lemma 4.1, we obtain

v(m+1)n−1 = vmn−1vn−1−Bnv(m−1)n−1 . (4.85)

Thus we have proved (4.80).

Next we shall prove (4.81). Whenm= 2, we can easily check (4.81) by (4.80). We assume

thatvmn−1 satisfy (4.81) for a certainm≥ 2. Then we shall show thatv(m+1)n−1 satisfy (4.81).

From (4.80) and the assumption, we have

v(m+1)n−1 = vmn−1vn−1−Bnv(m−1)n−1 (4.86)

= vn−1(Gm(vn−1)−BnGm−2(vn−1))−Bn(vn−1Gm−1(vn−1)−BnGm−3(vn−1))

= (vn−1Gm(vn−1)−BnGm−1(vn−1))−Bn(vn−1Gm−2(vn−1)−BnGm−3(vn−1)) .

SinceGm(z) satisfy

Gm+1(z) = zGm(z)−BnGm−1(z) , (4.87)

we obtain

v(m+1)n−1 = Gm+1(vn−1)−BnGm−1(vn−1) . (4.88)

By induction, we have proved (4.81).

In this paragraph, we finally derive the UvN hierarchy. We introduce the variablesu(m)n such

that

u(m)n := vmn−1 , m= 2,3,4, . . . , n = 0,1,2, . . . . (4.89)

Namely, we consider the map

u(m)0 = v0 7→ u(m)

1 = vm−1 7→ u(m)2 = vm2−1 7→ u(m)

3 = vm3−1 7→ · · · . (4.90)

55

By virtue of them-tuple addition formulas (4.81), we derive a hierarchy of nonautonomous

discrete dynamical systems

u(2)n+1 = (u(2)

n )2−2Bn , (4.91)

u(3)n+1 = (u(3)

n )3−3Bnu(3)n , (4.92)

u(4)n+1 = (u(4)

n )4−4Bn(u(4)n )2 +2B2n , (4.93)

u(5)n+1 = (u(5)

n )5−5Bn(u(5)n )3 +5B2nu(5)

n , (4.94)

and so on. We remark that determinantal solutions for systems (4.91)–(4.94) can be obtained

from (4.78) and (4.89). When we setB = 0 and replace the variableu(2)n such that

u(2)n → 1−2un , (4.95)

we derive a solvable logistic map

un+1 = 2un(1−un) , n = 0,1,2, . . . . (4.96)

The system (4.96) is not chaotic system for initial value0 < u0 < 1, and it converges to1/2

exponentially. Next we setB = 1 and replace the variablesu(m)n such that

u(m)n → 2(1−2u(m)

n ) . (4.97)

Then we obtain the UvN hierarchy from (4.91)–(4.94) by

u(2)n+1 = 4u(2)

n (1−u(2)n ) , (4.98)

u(3)n+1 = u(3)

n (3−4u(3)n )2 , (4.99)

u(4)n+1 = 16u(4)

n (1−u(4)n )(1−2u(4)

n )2 , (4.100)

u(5)n+1 = u(5)

n (5−4u(5)n (5−4u(5)

n ))2 , (4.101)

and so on. Furthermore we obtain the determinantal solutions for the UvN hierarchy by

u(m)n =

12− 1

4(Fmn−Fmn−2) , m= 2,3,4, . . . , n = 0,1,2, . . . , (4.102)

whereA = 2(1−2u(m)0 ), B = 1 andFn are defined by (4.79).

Relationship to the known determinantal solution [11] and the analytic solution [69] of the

logistic map is not clear. The determinants which appear in [11] look rather different fromF2n.

Indeed, the value of the parameterµ of the logistic mapun+1 = µ un(1−un) is not specified

in [11] and [69]. Recently the quadratic map (4.7) in real and complex domains is reviewed in

[54].

56

7.2. Lyapunov exponents.Let us restrict the initial valueu(m)0 to real values such that

0 < u(m)0 < 1. We shall compute the Lyapunov exponents of the UvN hierarchy without use

of explicit invariant measures. Let us writeu(m)n+1 = Ψ(u(m)

n ) for n = 0,1,2, . . .. The Lyapunov

exponents are expressed as

λ := limn→∞

1n

n−1

∑j=0

log∣∣∣Ψ′(u(m)

j )∣∣∣ . (4.103)

Here we consider the partial differentiation ofΨ(u(m)j ) with respect tou(m)

0 , which are given by

∂Ψ(u(m)j )

∂u(m)0

= Ψ′(u(m)j )

∂u(m)j

∂u(m)0

. (4.104)

From (4.104) andu(m)j = Ψ(u(m)

j−1), it follows that

Ψ′(u(m)j ) =

∂Ψ(u(m)j )

∂u(m)0

∂Ψ(u(m)j−1)

∂u(m)0

, j = 0,1,2, . . . ,n−1. (4.105)

From (4.105) andu(m)n = Ψ(u(m)

n−1), the Lyapunov exponents (4.103) are written as

λ = limn→∞

1n

log

∣∣∣∣∣∂u(m)

n

∂u(m)0

∣∣∣∣∣ . (4.106)

Inserting the solution (4.102) into (4.106), we obtain

λ = limn→∞

1n

log

∣∣∣∣∣∂ (Fmn−Fmn−2)

∂u(m)0

∣∣∣∣∣ . (4.107)

Using Lemma 4.4, we have

λ = limn→∞

1n

log|mnFmn−1|= logm+ limn→∞

1n

log|Fmn−1| . (4.108)

SinceFn satisfy the second order linear difference equation

Fn+1−2(1−u(m)0 )Fn +Fn−1 = 0, (4.109)

the determinantsFn can be also expressed as

Fn = c1 r1n +c2 r2

n . (4.110)

Herer1, r2 are the roots of the characteristic equation

x2−2(1−u(m)0 )x+1 = 0, (4.111)

57

andc1, c2 are determined by the initial condition. From the condition0 < u(m)0 < 1, it follows

that |r1| = |r2| = 1. Thus there exists a positive constantM such that0≤ |Fn| < M. Moreover

we remove the zeros ofFmn−1(u(m)0 ) = 0 from the initial region0 < u(m)

0 < 1, then we have

0 < |Fn|< M. Therefore we obtainL0 < log|Fmn−1|< L1, whereL0 andL1 are certain positive

constants. It is concluded that

limn→∞

1n

log|Fmn−1|= 0. (4.112)

We can state:

THEOREM 4.3. Let us restrict the initial valuesu(m)0 to real values such that0 < u(m)

0 < 1.

Then the Lyapunov exponents of the UvN hierarchy arelogm.

8. Determinantal solutions for hierarchy of the Steffensen iteration

In this section, we give determinantal solutions for the discrete dynamical systems corre-

sponding to the extended Steffensen method which is proposed in Chapter 3.

Let us consider the quadratic equation

f (z) := z2 +2bz+c = 0, (4.113)

wherez is a complex variable, andb, c are some complex constants. Rearranging Eq. (4.113),

we have the equation,

z=(a−b)z−cz+(a+b)

=: φ(z) , (4.114)

wherea is an auxiliary and arbitrary constant. We write the right hand side of Eq. (4.114)

asφ(z). We consider the extended Steffensen method (3.18) for Eq. (4.113) with the simple

iteration functionφ(z). The hierarchy of the Steffensen type iterations are given by

u(m)n+1 = Φm−1(u

(m)n ) , m= 2,3,4, . . . , n = 0,1,2, . . . , (4.115)

from (3.18). In [6, 7], Arai, Okamoto, and Kametaka find a new addition formula forcot(x) in

terms of addition formulas for a three parameter family of functions. The aim of this section is

to obtain determinantal solutions for the hierarchy (4.115) by using a theorem in [6, 7].

To find solutions, we begin to consider the simple iteration

vn+1 = φ(vn) =(a−b)vn−cvn +(a+b)

, j = 0,1,2, . . . . (4.116)

We present addition formulas ofvn, which are resulted from the following theorem.

58

THEOREM 4.4 (Addition formula). Let the auxiliary parametera be set bya = b+v0. The

sequencevn generated by the iteration(4.116)satisfies them-tuple addition formulas

vm(n+m−1)−1 =

m︷︸︸︷∣∣∣∣∣∣∣∣∣∣

vn−1 vn · · · vn+m−2

vn vn+1 · · · vn+m−1...

......

vn+m−2 vn+m−1 · · · vn+2m−3

∣∣∣∣∣∣∣∣∣∣

/

m+1︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1 · · · 1

1 vn−1 vn · · · vn+m−2

1 vn vn+1 · · · vn+m−1...

......

...

1 vn+m−2 vn+m−1 · · · vn+2m−3

∣∣∣∣∣∣∣∣∣∣∣∣∣

,

(4.117)

for m= 2,3,4, . . . andn = 1,2,3, . . ..

Proof. In order to prove it, we first introduce the theorem in the literature [6, 7]. Let p(z)be the three parameter family of functions defined by

p(z) := p(α,β ,γ;z) =α γz−βγz−1

(4.118)

with α−β , 0, γ , 0. Then functionsp(z) satisfies the addition formula

p(x1 +x2 + · · ·+xm+y1 +y2 + · · ·+ym) =

∣∣∣∣∣∣∣∣∣∣

p(x1+y1) p(x1+y2) · · · p(x1+ym)p(x2+y1) p(x2+y2) · · · p(x2+ym)

......

...

p(xm+y1) p(xm+y2) · · · p(xm+ym)

∣∣∣∣∣∣∣∣∣∣

/

∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1 · · · 1

1 p(x1+y1) p(x1+y2) · · · p(x1+ym)1 p(x2+y1) p(x2+y2) · · · p(x2+ym)...

......

...

1 p(xm+y1) p(xm+y2) · · · p(xm+ym)

∣∣∣∣∣∣∣∣∣∣∣∣∣

,

(4.119)

provided thatγxi+y j , 1.

We remark thatp(z) satisfies the relations

p(z+1) =(αγ−β ) p(z)−αβ (γ−1)

(γ−1) p(z)+(α−βγ), (4.120)

p(x+y) =p(x)p(y)−αβ

p(x)+ p(y)− (α +β ). (4.121)

In order to adapt the addition formula (4.119) tovn, we compare the recurrence relations (4.116)

with (4.120). From the comparison, we set the parameters as

α +β =−2b, αβ = c, γ =2a+α−β2a−α +β

. (4.122)

59

Thus we obtainvn = p(n+n0), where the integern0 is determined by an initial condition. Here

we choosen0 = 1, namely,v0 = p(1). Then we should restrict the auxiliary parametera as

a := b+v0 . (4.123)

Insertingxi = n+ i − 1, y j = j − 1 into (4.119), we therefore obtain the addition formulas

(4.117).

Next we assume that there exists a natural numberl for which u(m)n = vl−1 holds for each

stepn. From Theorem 4.4 and the assumption, the iteration (4.115) is rewritten as

u(m)n+1 = Φm−1(u

(m)n ) = Φm−1(vl−1) = vm(l+m−1)−1 (4.124)

for each stepn. We shall determine a natural numberl for each stepn. Startingu(m)0 = v0 and

l = 1, and computing the relation (4.124) recursively, we have

u(m)1 = Φm−1(u

(m)0 ) = Φm−1(v0) = vm2−1 , if l = 1, (4.125)

u(m)2 = Φm−1(u

(m)1 ) = Φm−1(vm2−1) = vm3+m2−m−1 , if l = m2 , (4.126)

u(m)3 = Φm−1(u

(m)2 ) = Φm−1(vm3+m2−m−1) = vm4+m3−m−1 , if l = m3 +m2−m, (4.127)

and so on. By induction, we obtain

u(m)n = vmn+1+mn−m−1 = v(m+1)(mn−1) , n = 0,1,2, . . . . (4.128)

The relation (4.128) yields the map

u(m)0 = v0 7→ u(m)

1 = v(m+1)(m−1) 7→ u(m)2 = v(m+1)(m2−1) 7→ · · · (4.129)

which is the extended Steffensen iteration.

We finally obtain determinantal solution. Since the recurrence relation (4.116) is a discrete

Riccati equation (4.41), the determinantal solution forvn can be obtained from (4.44) and (4.45).

By virtue of (4.128), we finally obtain the determinantal solution for (4.115) by

u(m)n = u(m)

0 −BF(m+1)(mn−1)−1

F(m+1)(mn−1), n = 0,1,2, . . . , (4.130)

60

whereA = f ′(u(m)0 ), B = f (u(m)

0 ) and

F−1 = 0, F0 = 1, Fn =

n︷︸︸︷∣∣∣∣∣∣∣∣∣∣∣∣∣

A B

1 A B. . .

. . .. . .

1 A B

1 A

∣∣∣∣∣∣∣∣∣∣∣∣∣

, n = 1,2,3, . . . . (4.131)

It should be noted that the determinantal solution (4.130)–(4.131) is also expressed as the

continued fraction

u(m)n = u(m)

0 − B

A− B

A· · ·− B

A.

︸︷︷︸(m+1)(mn−1)

(4.132)

We have constructed the determinantal solution (4.130)–(4.131) by only using four arith-

metic operations. Here we ease this restriction. Let us allow to use the operation of square root.

Another type solution for (4.115) is obtained by

u(m)n = p

(λ1, λ2,

u(m)0 −λ2

u(m)0 −λ1

; mn+1 +mn−m

)(4.133)

from (4.118), (4.122), (4.128) andvn = p(n+1). Hereλ1 andλ2 denote the roots of the equation

f (z) = 0.


In this chapter, we have obtained the determinantal solutions for irreversible discrete equa-

tions. We have dealt with the hierarchy of the UvN system, and the hierarchies of discrete dy-

namical systems which are derived by applying the Newton type iterations and the Steffensen

type iterations to a quadratic equation. According to the setting of parameters and initial condi-

tions, these systems give rise to algorithms having higher order convergence rates, or solvable

chaotic systems. For all cases, we have constructed the explicit solutions in a unified way.

Firstly, we have obtained the determinantal solutionsvn for the second order linear differ-

ence equation and the discrete Riccati equation. We have derived the addition formulas for the

solutionsvn (Theorems 4.1, 4.2, 4.4). At the next step, we have focused only on the valuesvmn

for integersm≥ 2. Then we have introduced the new variablesu(m)n = vmn for eachm. Finally,

we have showed that the addition formulas yield the irreversible dynamical systems ofu(m)n . As

61

a result, we have derived the hierarchies of new solvable irreversible dynamical systems and

have obtained their determinantal solutions simultaneously.

From the determinantal solutions for the UvN hierarchy, we have obtained the Lyapunov

exponents of them without explicit use of invariant measures (Theorem 4.3).

62

CHAPTER 5

Concluding Remarks

In this thesis, we have studied integrability of a continuous evolution equation and some

discrete equations. As an application of the soliton theory, we have proposed a numerical

algorithm based on the techniques in the nonlinear integrable systems.

In Chapter 2, we have considered the GDNLS equation. We first have constructed the trav-

eling wave solution which is valid for any real values of parameters. We have applied the

Painleve test to the GDNLS equation for detecting integrability. We have shown that the equa-

tion possesses the Painleve property in a strict sense only for the known integrable cases of

parameters. Therefore we have shown that it possesses a conditional Painleve property for an

infinite number of cases of conditions for parameters, which is the same condition as that of the

single-valued property of the traveling wave solution. When the GDNLS equation has the con-

ditional Painleve property, it is necessary for the functionφ(x, t) to satisfy an equation which is

transformed to the dispersionless KdV equation. We remark the interesting fact that the same

condition forφ(x, t) appeared at the Painleve analysis of the long and short wave interaction

equation by Yoshinaga [87, 88]. Next we have examined stability of the solitary wave by the

numerical simulation. Remarkable difference between integrable case and non-integrable case

has not been observed, except for the quantities of ripples generated by interactions. The travel-

ing wave solution is stable in interactions and behaves like a soliton. In conclusion, the GDNLS

equation is a near-integrable system which has a conditional Painleve property and a stable

soliton-like traveling wave solution. Further theoretical analysis on stability may be necessary.

In Chapter 3, we have proposed an extension of the Steffensen iteration for finding a rootαof the nonlinear equationx= φ(x). We have developed the extended Steffensen method in terms

of thek-th Shanks transform which is a sequence convergence acceleration algorithm. The re-

sulting iteration method does not need any derivative. And it has a higher order convergence

rate, although the Shanks transform is originally a linearly convergent algorithm. If the equa-

tion satisfiesφ ′(α) , 0,±1, then the sequence generated by the extended Steffensen method

has the(k+1)-th order convergence rate (Theorem 3.1). On the other hand, if the equation sat-

isfiesφ ′(α) = 0, then the extended Steffensen iteration has remarkably the(k+2)2k−1-th order

63

convergence rate (Theorem 3.2). These theoretical convergence rates have been verified in nu-

merical examples (Examples 1, 2). For the implementation of the extended Steffensen iteration,

theε-algorithm is especially useful to decrease the amount of computations in the calculation of

Hankel determinants. This algorithm is stable for errors and equivalent to the discrete potential

KdV equation. Consequently, computation due to the numbers of mappings takes a major part

of the computational complexity. We have shown that the extended Steffensen iteration with

k = 2 has the minimal numbers of mappings in a special case of the Kepler equation (Example

3). Moreover, the extended Steffensen iteration converges for more cases of parameters than

the Newton method.

In Chapter 4, we have obtained the determinantal solutions for irreversible discrete equa-

tions. We have dealt with the hierarchy of the UvN system, and the hierarchies of discrete

dynamical systems which are derived by applying the Newton type iterations and the Stef-

fensen type iterations to a quadratic equation. According to the setting of parameters and initial

conditions, these systems give rise to algorithms having higher order convergence rates, or solv-

able chaotic systems. For all cases, we have constructed the solutions in a unified way. Firstly,

we have obtained the determinantal solutionsvn for some linear systems. We have derived the

addition formulas for the solutionsvn. At the next step, we have focused only on the valuesvmn

for integersm≥ 2. Then we have introduced the new variablesu(m)n = vmn for eachm. Finally,

we have showed that the addition formulas yield the irreversible dynamical systems ofu(m)n . As

a result, we have obtained the hierarchies of new solvable irreversible dynamical systems and

their determinantal solutions simultaneously.

64

Acknowledgments

The author firstly would like to thank all people who helped him with this thesis.

The author would like to express his deepest gratitude to his supervisor, Professor Y. Naka-

mura, for continuous encouragements and many invaluable instructions. He also acknowledge

Professors M. Ohmiya, Y. Watanabe, and K. Kajiwara of Doshisha University for valuable ad-

vises and encouragements. He also thanks Dr. S. Tsujimoto and Dr. A. Nagai for many fruitful

discussions and helpful advises. The author is grateful to whole members of Nakamura Labo-

ratory, Applied Mathematics Laboratory of Doshisha University, and Suuri Kyoshitsu of Osaka

University for useful discussions and advises that enriched his study.

The author thanks Dr. T. Yoshinaga and Dr. H. Harada for useful comments and discussions

about the Painleve analysis. Thanks are also due to Mr. M. Deguchi and Mr. Y. Onoda for

assistance in numerical simulations of the GDNLS equation. He also thanks Dr. K. Umeno

for the fascinating lecture about solvable chaotic systems. The author would like to thank to

Professors T. Yamamoto, Y. Kametaka, H. Nagai, N. Osada, and H. Yoshida for giving him

helpful advises and suggestions.

The author finally would like to express his sincere gratitude to his parents.

65

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List of Authors Papers Cited in the Thesis and Related Works

Original Papers

[1] Koichi Kondo, Kenji Kajiwara, and Kai Matsui, Solution and integrability of a gener-

alized derivative nonlinear Schrodinger equation, J. Phy. Soc. Japan66 (1997), 60–66.

(Chapter 2)

[2] Koichi Kondo and Yoshimasa Nakamura, An extension of the Steffensen iteration and

its computational complexity, Interdisciplinary Information Sciences4 (1998), 129–

138. (Chapter 3)

[3] Koichi Kondo and Yoshimasa Nakamura, Determinantal solutions of solvable chaotic

systems, preprint. (Chapter 4)

Proceedings Papers

[1] Koichi Kondo and Yoshimasa Nakamura, Newton-Steffensen-Shanks, RIMS Kokyuroku,

Kyoto Univ.

近藤弘一,中村佳正,ニュートン・ステファンセン・シャンクス,京都大学数理解

析研究所講究録 1020(1997), 28–38.

[2] Koichi Kondo and Yoshimasa Nakamura, A new iteration method of the Steffensen

type having higher order convergence rate, RIMS Kokyuroku, Kyoto Univ.

近藤弘一,中村佳正,高次収束する Steffensen型反復解法,京都大学数理解析研究

所講究録 1040(1998), 228–236.

[3] Koichi Kondo and Yoshimasa Nakamura, Determinantal solutions for the Newton

method and solvable chaotic systems, RIAM Kokyuroku, Kyushu Univ.

近藤弘一,中村佳正,ニュートン法と可解カオスの行列式解,九州大学応用力学研

究所研究集会報告 11ME-S4(2000), 53–58.

[4] Satoshi Tsujimoto and Koichi Kondo, Molecule solutions of discrete soliton equations

and orthogonal polynomials, RIMS Kokyuroku, Kyoto Univ.

辻本諭,近藤弘一,離散方程式の分子解と直交多項式,京都大学数理解析研究所講

究録 1170(2000), 1–8.

71

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